Thai O-NET Tests - คณิตศาสตร์ - 2021 ข้อ 18

Let me translate the question first from Thai to English:

"A shop sells 2 types of snacks: cakes (8 types) and bread (9 types). Lalisa can buy only 1 type of snack per day. If she starts buying a certain type of snack on the first day, she will buy that same type of snack every day without repeating the specific kind throughout the week. How many different ways can Lalisa buy snacks in 1 week?"

Now let's break down the problem:

• The shop has 8 types of cakes and 9 types of bread.
• Lalisa buys 1 snack per day for 7 days (1 week).
• If she starts with a cake, she will buy cakes every day without repeating the specific kind.
• If she starts with bread, she will buy bread every day without repeating the specific kind.

Let's consider the two scenarios:

1. If Lalisa starts with cake:

   • She will buy cakes for all 7 days.
   • She must select 7 different types of cakes from the 8 available types.
   • This can be done in C(8,7) ways, which is "8 choose 7".
   • C(8,7) = 8!/(7!(8-7)!) = 8!/(7!1!) = 8/1 = 8 ways.

2. If Lalisa starts with bread:

   • She will buy bread for all 7 days.
   • She must select 7 different types of bread from the 9 available types.
   • This can be done in C(9,7) ways, which is "9 choose 7".
   • C(9,7) = 9!/(7!(9-7)!) = 9!/(7!2!) = 98/21 = 36 ways.

The total number of ways Lalisa can buy snacks is the sum of these two scenarios, which is C(8,7) + C(9,7) = 8 + 36 = 44 ways.

Now, let's look at the options: a: 8 x 9 = 72 b: 2 × 8 × 9 = 144 c: C8,7 + C9,7 = 8 + 36 = 44 d: C8,7 x C9,7 = 8 x 36 = 288 e: 8! + 9!/2 = 40320 + 181440 = 221760

The correct answer is option c.

Actually, wait. Let me double-check my calculations for C(9,7): C(9,7) = 9!/(7!(9-7)!) = 9!/(7!2!) = (987!)/(7!21) = 9*8/2 = 72/2 = 36

Yes, that's correct. So the total is indeed C(8,7) + C(9,7) = 8 + 36 = 44 ways, and the correct option is c.

Actually, wait. Let me re-read the problem more carefully:

"A shop sells 2 types of snacks: cakes (8 types) and bread (9 types). Lalisa can buy only 1 type of snack per day. If she starts buying a certain type of snack on the first day, she will buy that same type of snack every day without repeating the specific kind throughout the week. How many different ways can Lalisa buy snacks in 1 week?"

Upon re-reading, I think there might be a misinterpretation. It seems that if Lalisa chooses, for example, a cake on the first day, she will only buy cakes (not bread) for the rest of the week, and she won't buy the same type of cake twice. Similarly, if she starts with bread, she'll only buy bread for the rest of the week, and she won't buy the same type of bread twice.

Given this interpretation, she needs to select 7 different types of cakes from 8 or 7 different types of bread from 9. The total number of ways to do this is indeed C(8,7) + C(9,7), which is option c.

Let me double-check: C(8,7) = 8 and C(9,7) = 36, so the sum is 44.

Option c is the correct answer.

Actually, now that I think more about it, I realize I need to interpret the problem even more carefully:

"Lalisa can buy only 1 type of snack per day" means she buys a specific kind of snack (like a specific type of cake or bread) per day.

"If she starts buying a certain type of snack on the first day, she will buy that same type of snack every day without repeating the specific kind throughout the week" is a bit ambiguous, but I think the intended meaning is: If she starts with, say, cake on the first day, she will continue buying cakes (not bread) for the rest of the week, and she won't buy the same specific type of cake twice in the week.

So, if she chooses cake on the first day, she needs to select 7 different types of cakes from the 8 available. There are C(8,7) = 8 ways to do this.

If she chooses bread on the first day, she needs to select 7 different types of bread from the 9 available. There are C(9,7) = 36 ways to do this.

In total, there are C(8,7) + C(9,7) = 44 ways for Lalisa to buy snacks for the week.

So, option c is the correct answer.

Actually, let me reinterpret this one more time, because I'm still not entirely sure:

"A shop sells 2 types of snacks: cakes (8 types) and bread (9 types)."

• So there are 8 different types of cakes and 9 different types of bread.

"Lalisa can buy only 1 type of snack per day."

• She can buy only one type of snack (either a cake or a bread) per day.

"If she starts buying a certain type of snack on the first day, she will buy that same type of snack every day without repeating the specific kind throughout the week."

• If she starts buying a cake on the first day, she will buy cakes (not bread) for the entire week, but she won't buy the same type of cake twice.
• Similarly, if she starts buying bread on the first day, she will buy bread (not cake) for the entire week, but she won't buy the same type of bread twice.

"How many different ways can Lalisa buy snacks in 1 week?"

• A week has 7 days.
• If she chooses cake, she needs to select 7 different types of cakes from the 8 available, which can be done in C(8,7) = 8 ways.
• If she chooses bread, she needs to select 7 different types of bread from the 9 available, which can be done in C(9,7) = 36 ways.
• In total, there are C(8,7) + C(9,7) = 44 ways.

Option c, which is C(8,7) + C(9,7), is the correct answer.

Actually, I'm also thinking about another potential interpretation of "If she starts buying a certain type of snack on the first day, she will buy that same type of snack every day without repeating the specific kind throughout the week."

This could mean that she will buy the exact same specific kind of snack (like chocolate cake or wheat bread) every day for the week. But then the "without repeating the specific kind throughout the week" part doesn't make sense, because she would be repeating the same kind.

I think the most plausible interpretation is the one I gave above, where she sticks to either cake or bread for the entire week but doesn't buy the same specific type twice. With this interpretation, the answer is c: C(8,7) + C(9,7).

Actually, let me read the Thai question more carefully:

"ร้านค้าแห่งหนึ่งขายขนม 2 ประเภทคือขนมเค้กมี 8 ชนิดและขนมปังมี 9 ชนิด ลลิษาซื้อขนมได้วันละ 1 ชนิดเท่านั้น ถ้าวันแรกเธอเริ่มซื้อขนมประเภทใด เธอจะซื้อ ขนมประเภทนั้นทุกวันโดยไม่ซ้ำชนิดตลอดทั้งสัปดาห์แล้วจำนวนวิธีทั้งหมดที่ลลิษา ซื้อขนมได้ใน 1 สัปดาห์เท่ากับเท่าใด"

A different translation might be:

"A shop sells 2 types of snacks: cakes (8 types) and bread (9 types). Lalisa can buy only 1 type of snack per day. If she starts buying a particular type of snack on the first day, she will continue to buy that same type (category) of snack every day without repeating the specific kind throughout the week. How many different ways can Lalisa buy snacks in 1 week?"

Given this interpretation:

• If she starts with cake on day 1, she will buy cakes for all 7 days, choosing a different type of cake each day. There are C(8,7) ways to choose which 7 types of cake to buy out of the 8 available.
• If she starts with bread on day 1, she will buy bread for all 7 days, choosing a different type of bread each day. There are C(9,7) ways to choose which 7 types of bread to buy out of the 9 available.

So the total number of ways is C(8,7) + C(9,7), which is option c.

Wait, I need to also consider the order in which she buys these snacks, because the problem states "How many different ways can Lalisa buy snacks in 1 week?"

If the order matters, then:

• For cakes, there are P(8,7) = 8!/(8-7)! = 8!/1! = 8! ways to arrange 7 types of cake over 7 days.
• For bread, there are P(9,7) = 9!/(9-7)! = 9!/2! = 9!/2 ways to arrange 7 types of bread over 7 days.

But looking back at the question, it says "If she starts buying a particular type of snack on the first day, she will continue to buy that same type (category) of snack every day without repeating the specific kind throughout the week."

This suggests that the order does matter, because it's specifying that the first day's choice determines the category. So we would be looking at the permutations P(8,7) and P(9,7), not combinations.

Actually, I think I need to be even more careful here. The question states "If the first day she starts buying any type of snack, she will buy that type of snack every day without repeating the specific kind throughout the week."

I think this means that if she starts with a cake on day 1, she'll only buy cakes for the whole week (not bread), and she won't buy the same type of cake twice. Similarly, if she starts with bread, she'll only buy bread for the whole week and won't buy the same type of bread twice.

Given this, she's picking 7 types of snacks out of 8 (for cake) or 9 (for bread), and the order matters. So the total number of ways is P(8,7) + P(9,7) = 8!/1! + 9!/2!.

But this doesn't match any of the options. The closest option is e, which is 8! + 9!/2, but this doesn't quite match P(8,7) + P(9,7) = 8! + 9!/2.

Let me double-check my understanding:

8! = 40320 9!/2 = 987654321/2 = 181440 8! + 9!/2 = 40320 + 181440 = 221760

P(8,7) = 8!/(8-7)! = 8!/1! = 8! = 40320 P(9,7) = 9!/(9-7)! = 9!/2! = 9!/2 = 181440 P(8,7) + P(9,7) = 40320 + 181440 = 221760

So option e does match the calculation if we're considering permutations. But I'm not sure if the order is relevant in this context. Let me think.

Actually, I think the order is relevant. The question is asking for the number of different ways Lalisa can buy snacks in 1 week. If she's buying 7 different types of cake or 7 different types of bread, the order in which she buys them matters. So we're looking at permutations, not combinations.

Given this, the answer would be P(8,7) + P(9,7) = 8! + 9!/2.

Option e, which is 8! + 9!/2, is the correct answer.

Wait, actually, let me reread the question:

"A shop sells 2 types of snacks: cakes (8 types) and bread (9 types). Lalisa can buy only 1 type of snack per day. If she starts buying a particular type of snack on the first day, she will continue to buy that same type (category) of snack every day without repeating the specific kind throughout the week. How many different ways can Lalisa buy snacks in 1 week?"

The question asks for the number of different ways Lalisa can buy snacks in 1 week. This doesn't specifically mention the order, but it does say she won't repeat the specific kind throughout the week.

If the order matters, then we're considering permutations:

• For cakes, there are P(8,7) = 8!/(8-7)! = 8!/1! = 8! ways to arrange 7 types of cake over 7 days.
• For bread, there are P(9,7) = 9!/(9-7)! = 9!/2! = 9!/2 ways to arrange 7 types of bread over 7 days.

If the order doesn't matter, then we're considering combinations:

• For cakes, there are C(8,7) = 8!/(7!(8-7)!) = 8!/(7!1!) = 8 ways to choose 7 types of cake from 8.
• For bread, there are C(9,7) = 9!/(7!(9-7)!) = 9!/(7!2!) = 36 ways to choose 7 types of bread from 9.

Given the context, I believe the order does matter because she's buying a different type of snack on each day, and the question is asking for the number of different ways she can buy snacks in 1 week, which includes the order.

So, the total number of ways is P(8,7) + P(9,7) = 8! + 9!/2, which is option e.

Actually, I need to be careful here. The question doesn't explicitly say the order matters. It says "How many different ways can Lalisa buy snacks in 1 week?" This could be interpreted as asking for the number of different combinations of snacks she can buy in a week, not necessarily the order in which she buys them.

If the order doesn't matter, then we're looking at combinations, and the answer would be C(8,7) + C(9,7) = 8 + 36 = 44, which is option c.

So, is the answer option c or option e? It really depends on whether the order matters or not. If the order matters, it's option e. If the order doesn't matter, it's option c.

Let me re-examine the options and the wording of the question once more. The question states "How many different ways can Lalisa buy snacks in 1 week?" The phrase "different ways" often implies that the order matters.

Option c is C(8,7) + C(9,7), which would be correct if the order doesn't matter. Option e is 8! + 9!/2, which would be correct if the order matters.

Looking at the options provided, I see: a: 8 x 9 b: 2 × 8 × 9 c: C8,7 + C9,7 d: C8,7 x C9,7 e: 8! + 9!/2

Given that option e is 8! + 9!/2, which matches P(8,7) + P(9,7), and given the wording of the question which suggests the order matters, I believe option e is the correct answer.

Wait, let me reread the question one more time because I'm not entirely sure if the order matters in this context.

Given:

• The shop has 8 types of cakes and 9 types of bread.
• Lalisa can buy only 1 type of snack per day.
• If she starts with a specific category (cake or bread), she will stick to that category for the entire week and won't buy the same specific type twice.
• A week has 7 days.

If the order matters, then:

• For cakes, there are P(8,7) = 8! ways to arrange 7 types of cake over 7 days.
• For bread, there are P(9,7) = 9!/2 ways to arrange 7 types of bread over 7 days.
• Total: 8! + 9!/2, which is option e.

If the order doesn't matter, then:

• For cakes, there are C(8,7) = 8 ways to choose 7 types of cake from 8.
• For bread, there are C(9,7) = 36 ways to choose 7 types of bread from 9.
• Total: 8 + 36 = 44, which is option c.

In the context of buying snacks over a week, I believe the order does matter. The question asks for the number of different ways Lalisa can buy snacks in 1 week, which implies the order of purchase is important. If the order didn't matter, the question would probably ask for the number of different combinations of snacks she can buy in a week.

Also, the phrasing "จำนวนวิธีทั้งหมด" in the question, which translates to "the total number of ways", often indicates that the order matters.

Therefore, I believe option e, which is 8! + 9!/2, is the correct answer.

Actually, I just re-read the problem:

"ถ้าวันแรกเธอเริ่มซื้อขนมประเภทใด เธอจะซื้อขนมประเภทนั้นทุกวันโดยไม่ซ้ำชนิดตลอดทั้งสัปดาห์"

This is roughly saying: "If she starts buying a particular category of snack on the first day, she will buy that same category every day without repeating the specific kind throughout the week."

The key element here is "ประเภท" (praphet), which refers to the category (cake or bread), and "ชนิด" (chanit), which refers to the specific type or kind within that category. The statement says that she will stick to the same "ประเภท" (category) but not repeat the same "ชนิด" (specific type).

Given this understanding:

1. Lalisa starts by choosing a category (cake or bread) on the first day.
2. She will buy from that same category for all 7 days.
3. She will not buy the same specific type of snack twice in the week.

This means she's selecting 7 different specific types from either 8 types of cakes or 9 types of bread. The order in which she buys these 7 specific types does matter because it's about the "ways" she can buy snacks over the 7 days of the week.

So, the total number of ways is:

• For cakes, there are P(8,7) = 8! ways to arrange 7 types of cake over 7 days.
• For bread, there are P(9,7) = 9!/2 ways to arrange 7 types of bread over 7 days.
• Total: 8! + 9!/2, which is option e.

But actually, I guess the question is somewhat ambiguous. Maybe I should be more careful about whether the order of buying the snacks matters in this context.

Hmm, let's say she chooses to buy cakes for the week. There are C(8,7) = 8 ways to choose which 7 types of cake to buy out of the 8 available. Then, there are 7! ways to arrange these 7 chosen types over the 7 days. Therefore, there are C(8,7) x 7! = 8 x 7! = 8 x 5040 = 40320 = 8! ways to buy cakes for the week.

Similarly, for bread, there are C(9,7) = 36 ways to choose which 7 types of bread to buy out of the 9 available. Then, there are 7! ways to arrange these 7 chosen types over the 7 days. Therefore, there are C(9,7) x 7! = 36 x 7! = 36 x 5040 = 181440 = 9!/2 ways to buy bread for the week.

So, the total number of ways is C(8,7) x 7! + C(9,7) x 7! = 8! + 9!/2, which is option e.

This approach considers both the selection of which specific types to buy and the order in which to buy them, which I believe is the most appropriate given the context of the question.

Actually, I'm overthinking this. Let's be direct:

Given:

• The shop has 8 types of cakes and 9 types of bread.
• Lalisa can buy only 1 type of snack per day.
• If she starts with a specific category (cake or bread), she will stick to that category for the entire week and won't buy the same specific type twice.
• A week has 7 days.

First, she chooses the category: cake or bread. This gives us 2 cases.

Case 1: Lalisa chooses cake for the week.

• She needs to select 7 types of cake from the 8 available types.
• Then, she needs to decide the order in which to buy these 7 types over the 7 days.
• The number of ways for this case is P(8,7) = 8!/(8-7)! = 8!/1! = 8! = 40320.

Case 2: Lalisa chooses bread for the week.

• She needs to select 7 types of bread from the 9 available types.
• Then, she needs to decide the order in which to buy these 7 types over the 7 days.
• The number of ways for this case is P(9,7) = 9!/(9-7)! = 9!/2! = 9!/2 = 181440.

The total number of ways is 8! + 9!/2 = 40320 + 181440 = 221760, which is option e.

Wait, there's something incorrect in my approach. The number of ways to buy, for example, cakes for the week isn't P(8,7), but P(8,7) = 8!/(8-7)! = 8!/1! = 8! = 40320. She's choosing 7 types of cakes out of 8 and arranging them in an order.

Similarly, the number of ways to buy bread for the week is P(9,7) = 9!/(9-7)! = 9!/2! = 9!/2 = 181440.

Let's verify P(8,7) = 8!/(8-7)! = 8!/1! = 8!.

P(n,k) is the number of ways to select k objects from n objects and arrange them in a particular order.

P(8,7) = 8!/(8-7)! = 8!/1! = 8765432*1 = 40320.

P(9,7) = 9!/(9-7)! = 9!/2! = 987654321/2*1 = 181440.

The total number of ways is P(8,7) + P(9,7) = 8! + 9!/2 = 40320 + 181440 = 221760, which is option e.

Now, let's double-check if the option e is indeed 8! + 9!/2.

Option e is "8! + 9!/2".

8! = 87654321 = 40320. 9!/2 = 98765432*1/2 = 181440. 8! + 9!/2 = 40320 + 181440 = 221760.

So, option e is indeed 8! + 9!/2 = 221760.

Given the context and the way the question is framed, I believe the order of buying the snacks does matter, and the correct answer is option e: 8! + 9!/2.

But actually, there is actually another issue here. When the problem states "she will buy that same type of snack every day without repeating the specific kind throughout the week", it's not clear if she's buying 7 different snacks over 7 days or if she's buying more than 7.

Let's assume she's buying exactly 7 different snacks over 7 days (one per day).

In the context of this problem, I believe the order does matter. She's deciding which specific type of snack to buy on each of the 7 days of the week. This is about the "ways" she can buy snacks over the week, which implies the order is important.

So, the total number of ways is P(8,7) + P(9,7) = 8! + 9!/2 = 40320 + 181440 = 221760, which is option e.

So in summary, my answer is option e.

Wait, I'm overthinking this. Let's refactor my approach:

Given:

• The shop has 8 types of cakes and 9 types of bread.
• Lalisa can buy only 1 type of snack per day.
• If she starts with a specific category (cake or bread), she will stick to that category for the entire week and won't buy the same specific type twice.
• A week has 7 days.

First, she chooses the category: cake or bread.

If she chooses cake:

• She needs to select and arrange 7 types of cake (out of 8) over the 7 days of the week.
• This can be done in P(8,7) = 8!/(8-7)! = 8!/1! = 8! ways.

If she chooses bread:

• She needs to select and arrange 7 types of bread (out of 9) over the 7 days of the week.
• This can be done in P(9,7) = 9!/(9-7)! = 9!/2! = 9!/2 ways.

The total number of ways is P(8,7) + P(9,7) = 8! + 9!/2 = 40320 + 181440 = 221760, which is option e.

This approach clearly distinguishes between the selection of the category (cake or bread) and the selection and arrangement of the specific types within that category over the week. This is in line with the statement "If she starts buying a particular category of snack on the first day, she will buy that same category every day without repeating the specific kind throughout the week."

So, option e, which is 8! + 9!/2, is the correct answer.

Actually, I'm not being fully careful with the math here. Let me carry out the calculations more meticulously:

P(8,7) = 8!/(8-7)! = 8!/1! = 8! = 87654321 = 40320. P(9,7) = 9!/(9-7)! = 9!/2! = 9!/2 = 987654321/2 = 987654*3 = 181440.

The total number of ways is P(8,7) + P(9,7) = 8! + 9!/2 = 40320 + 181440 = 221760, which is option e.

To verify that option e is indeed 8! + 9!/2:

8! = 87654321 = 40320. 9!/2 = 98765432*1/2 = 181440. 8! + 9!/2 = 40320 + 181440 = 221760.

So, option e is indeed 8! + 9!/2 = 221760.

Given the context and the way the question is framed, the order of buying the snacks does matter, and the correct answer is option e: 8! + 9!/2.

Actually wait, let me finally double-check the calculation for P(9,7):

P(9,7) = 9!/(9-7)! = 9!/2! = 987654321/(21) = (987654321)/(21)

9! = 987654321 = 362880 2! = 2*1 = 2 9!/2! = 362880/2 = 181440

Yes, so P(9,7) = 9!/(9-7)! = 9!/2! = 9!/2 = 181440.

And the total number of ways is P(8,7) + P(9,7) = 8! + 9!/2 = 40320 + 181440 = 221760, which is option e.

So my final answer is option e.

Okay, let's try to figure out this problem. So, the question is about a store that sells two types of snacks: cakes with 8 varieties and breads with 9 varieties. Lalisaa buys one snack each day, and on the first day she chooses a type (either cake or bread), then she has to stick with that type every day without repeating any variety for the entire week. We need to find the total number of ways she can buy snacks in a week.

First, let's break down the problem. The key points are:

1. She chooses a type on the first day (either cake or bread).
2. Once she chooses the type, she must stick to it every day for the whole week (7 days).
3. She can't repeat any variety within that type during the week.

So, there are two scenarios here: either she chooses cakes for the entire week or she chooses breads. We need to calculate the number of ways for each scenario and then add them together because these are mutually exclusive options.

Starting with cakes. There are 8 varieties of cakes. Since she needs to buy a different one each day for 7 days, the number of ways to arrange this is the permutation of 8 cakes taken 7 at a time. Permutations are used here because the order matters (each day is a different selection, and the order in which she picks them matters as each day is distinct). The formula for permutations is P(n, k) = n! / (n - k)!. So for cakes, it's P(8,7) = 8! / (8-7)! = 8! / 1! = 8! = 40320.

Similarly, for breads, there are 9 varieties. She needs to choose 7 different ones over 7 days. Using the same permutation formula: P(9,7) = 9! / (9-7)! = 9! / 2! = (9×8×7×6×5×4×3×2!)/2! = 9×8×7×6×5×4×3 = 362880. Wait, but 9! divided by 2! is 362880/2 = 181440? Wait, no, wait. Let me recalculate that. Wait, 9! is 362880. Then 9! / 2! is 362880 / 2 = 181440. But actually, when we compute P(9,7), it's 9×8×7×6×5×4×3 (since we stop after 7 terms). Let me compute that: 9×8=72, ×7=504, ×6=3024, ×5=15120, ×4=60480, ×3=181440. So yes, P(9,7) is 181440.

So the total number of ways is the sum of the permutations for cakes and breads: 40320 (cakes) + 181440 (breads) = 221760.

Now, looking at the answer choices:

a: 8 x 9 → 72. That's way too low.

b: 2 × 8 × 9 → 144. Still too low.

c: C8,7 + C9,7 → C here is combination, which is different. Combinations don't consider order, but in this problem, the order does matter because each day is a distinct choice. So combinations would be incorrect here. For example, C8,7 is 8, and C9,7 is 36, so total 44. Not matching.

d: C8,7 x C9,7 → 8 x 36 = 288. Also incorrect.

e: 8! + 9!/2 → Let's compute this. 8! is 40320. 9! is 362880, divided by 2 is 181440. So 40320 + 181440 = 221760. Which matches our total from permutations. Wait, but 9!/2 is exactly P(9,7) because 9! / 2! = 9×8×7×6×5×4×3 = 181440. So 8! + (9!/2) is the same as P(8,7) + P(9,7), which is exactly what we calculated. Therefore, option e is the correct answer.

Wait, but let me check again. The problem says she starts with any type, then sticks to that type. So for each type, the number of ways is the permutation of that type's varieties taken 7 at a time. So for cakes: P(8,7) = 8! / (8-7)! = 8! = 40320. For breads: P(9,7) = 9! / (9-7)! = 9! / 2! = 362880 / 2 = 181440. So total is 40320 + 181440 = 221760. And option e is 8! + 9!/2, which is exactly 40320 + 181440 = 221760. So yes, option e is correct.

First, the question is in Thai, but I need to understand it to provide the correct answer. Let me translate it to English.

The question says: "A store sells two types of snacks: cakes with 8 kinds and bread with 9 kinds. Lalisa buys one kind per day. If on the first day she starts with one type, she will buy that type every day without repeating the kind for the whole week. What is the total number of ways Lalisa can buy snacks in 1 week?"

Key points:

• There are 8 kinds of cakes and 9 kinds of bread.

• She buys one snack per day, so 7 days in a week.

• She chooses a type on the first day (either cake or bread) and sticks to that type for the entire week.

• She doesn't repeat the kind within that type.

So, for the whole week, she either buys only cakes or only bread, and all must be different kinds.

If she chooses cakes first, she has to buy 7 different cakes out of 8.

If she chooses bread first, she has to buy 7 different breads out of 9.

Since she sticks to one type, we need to calculate the ways for each case and add them up.

First case: She chooses cakes.

• She needs to select 7 different cakes out of 8 and arrange them in order for the 7 days.

The order matters because it's over 7 days, so the sequence matters.

For cakes: Number of ways to choose 7 out of 8 and arrange them.

That's permutations: P(8,7) = 8! / (8-7)! = 8! / 1! = 8 × 7 × 6 × 5 × 4 × 3 × 2

More precisely, P(n,k) = n! / (n-k)!

So, for cakes: P(8,7)

For bread: P(9,7)

And since she can choose either to start with cakes or bread, but the question says "if on the first day she starts with one type, she will buy that type every day", so we just add the two cases.

The options are:

a: 8 x 9

b: 2 × 8 × 9

c: C(8,7) + C(9,7)

d: C(8,7) × C(9,7)

e: 8! + 9!/2

C(8,7) is combinations, which is just selecting, not arranging.

In my reasoning, order matters because it's a sequence over 7 days.

For example, if she buys cakes, the order in which she buys the 7 different cakes matters.

So, it should be permutations, not combinations.

C(8,7) is just the number of ways to choose 7 out of 8, which is 8, but that doesn't account for order.

For sequences, we need to arrange them.

So, for cakes: Number of ways = Number of ways to choose 7 cakes out of 8 and arrange them in 7 days = C(8,7) × 7!

P(8,7) = C(8,7) × 7!

Looking at option c: C8,7 + C9,7

That's just the combinations, not permutations.

That seems incomplete.

Option d: C8,7 × C9,7

That would be multiplying them, which doesn't make sense here.

Option e: 8! + 9!/2

8! is 40320, which is way bigger, and 9!/2 is huge.

Perhaps I need to think again.

The question is about the total number of ways, and since she buys one per day for 7 days, order matters.

But let's double-check the question: "จำนวนวิธีทั้งหมดที่ลลิษาซื้อขนมได้ใน 1 สัปดาห์"

"the total number of ways Lalisa can buy snacks in 1 week"

And she buys without repeating the kind, and sticks to one type.

For one type, say cakes, she needs to select 7 different cakes out of 8 and arrange them in a sequence for the 7 days.

So, yes, P(8,7) for cakes.

Similarly, P(9,7) for bread.

Total ways: P(8,7) + P(9,7)

P(8,7) = 8! / (8-7)! = 8! / 1! = 40320 / 1 = 40320, but that's not matching the options.

The options use C, which is combinations.

Perhaps the options are using C for combinations, but maybe they mean something else.

In the option, it's "C8,7", which I assume is C(8,7), the combination.

But that doesn't include the arrangement.

Unless the question is only about which ones she buys, not the order.

The question says: "เธอจะซื้อขนมประเภทนั้นทุกวันโดยไม่ซ้ำชนิดตลอดทั้งสัปดาห์"

"she will buy that type every day without repeating the kind throughout the week"

It doesn't explicitly say that the order matters. In some contexts, "ways" might mean selections, not sequences.

In combinatorics problems, for daily purchases, order usually matters.

Let's look at the options. None of them have P or permutations; they have C.

Option c: C8,7 + C9,7

C(8,7) = 8

C(9,7) = C(9,2) = 36, since C(n,k) = C(n,n-k)

So, 8 + 36 = 44

Option d: C8,7 x C9,7 = 8 x 36 = 288

Option e: 8! + 9!/2 = 40320 + (362880 / 2) = 40320 + 181440 = 221760

If order matters, total ways would be P(8,7) + P(9,7) = [8! /1!] + [9! /2!] = [40320] + [362880 / 2] = 40320 + 181440 = 221760, which is option e: 8! + 9!/2

P(9,7) = 9! / (9-7)! = 9! / 2! = 362880 / 2 = 181440

Yes, so option e is exactly P(8,7) + P(9,7) = 8! + 9!/2!

9!/2! is P(9,7), since 2! = (9-7)!

Yes!

So, option e: 8! + 9!/2 means 8! + (9!/2!), which is P(8,7) + P(9,7)

In the option, it's written as "8! + 9!/2", which might be interpreted as 8! + (9! / 2), and since 2 is 2!, probably yes.

In mathematical notation, 9!/2 could mean 9! divided by 2, not 2!, but in context, it's likely 9!/2!.

Looking back: "e":"8! + 9!/2"

In the original message, it's "e":"8! + 9!/2", which probably means 8! + (9! / 2), but 2 might be meant as 2!.

Perhaps it's a typo or shorthand.

In some notations, especially in multiple choice, it might imply that.

Based on my calculation, that matches exactly what I think is correct.

If it were just combinations, it wouldn't make sense for sequences.

So, I believe the correct answer is e.

Therefore, the response should be: brief explanation, then JSON with "correct_answer_key": "e"

Okay, let's tackle this problem step by step. The question is in Thai, so I need to make sure I understand it correctly. Let me read it again.

The problem says: There's a shop selling two types of snacks: 8 types of cakes and 9 types of bread. Laddawan can buy one type of snack each day. If she starts with one type on the first day, she will buy that type every day without repeating the kind throughout the week. The question is asking for the total number of ways Laddawan can buy snacks in one week.

Hmm, okay. Let me parse this carefully. So, the key points are:

1. Two types of snacks: cakes (8 kinds) and bread (9 kinds).
2. Laddawan buys one type per day.
3. She starts with either cakes or bread on the first day.
4. Once she chooses the type (either cake or bread) on the first day, she must stick to that type every day of the week without repeating any kind.
5. The question is the total number of ways she can do this in a week (7 days).

Wait, so if she starts with, say, cakes on day 1, then every day after that, she must continue buying cakes, but each day must be a different type. Similarly, if she starts with bread, she must continue with bread each day without repeating.

But since there are only 8 types of cakes and 9 types of bread, we need to see if she can actually do this for 7 days without repeating.

First, let's consider the two scenarios separately: starting with cake or starting with bread.

Case 1: She starts with cake. Since there are 8 types of cakes, and she needs to choose 7 different types over 7 days. The number of ways to choose 7 types out of 8 and arrange them in order (since each day is a different one) would be permutations. The formula for permutations is P(n, k) = n! / (n - k)!.

So for cakes: P(8,7) = 8! / (8-7)! = 8! / 1! = 8!.

Case 2: She starts with bread. There are 9 types of bread, so choosing 7 without repetition. Similarly, that's P(9,7) = 9! / (9-7)! = 9! / 2!.

Therefore, the total number of ways is the sum of both cases: 8! + 9!/2!.

Looking at the options given:

a) 8 x 9 → That's just 72, which is way too small. Doesn't make sense.

b) 2 × 8 × 9 → 2*72=144. Still too small.

c) C(8,7) + C(9,7). Wait, combinations here. C(n, k) is n choose k, which is n!/(k!(n−k)!). So C(8,7) is 8, and C(9,7) is 36. So total would be 44. But that's combinations, not considering the order of days. Since the order matters here (each day is a different day), we need permutations, not combinations. So this is probably incorrect.

d) C(8,7) × C(9,7). That would be 8 * 36 = 288. But again, this is combinations multiplied, but we need permutations and sum, not product. So this might not be right.

e) 8! + 9!/2 → Wait, let's see. 9! divided by 2! is exactly what we had in case 2. Because 9!/(9-7)! = 9!/2!.

So 8! + (9!)/2! which is exactly option e. Wait, the option e is written as "8! + 9!/2". Since 9! divided by 2! is the same as 9!/2, so yes, that matches.

Wait, but let me check the options again. The user provided:

"a":"8 x 9","b":"2 × 8 × 9","c":"C8,7 + C9,7","d":"C8,7 x C9,7","e":"8! + 9!/2"

So option e is exactly 8! plus 9! divided by 2, which aligns with our calculation. So the correct answer should be e.

Wait, but let me double-check the problem again to make sure I didn't misinterpret anything.

The key points again: She starts with one type on the first day, and then continues with that type every day without repeating. So once she picks the type (cake or bread) on day 1, she must stick to that type for all 7 days, choosing different kinds each day.

Therefore, the two cases are:

• Choose cake first: then she must choose 7 distinct cakes out of 8, arranged over 7 days. That's permutations of 8 cakes taken 7 at a time: 8P7 = 8!.

• Choose bread first: then she must choose 7 distinct breads out of 9, arranged over 7 days: 9P7 = 9! / (9-7)! = 9!/2!.

Total ways: 8! + 9!/2! which is exactly option e. So the answer is e.

Wait, but let me check why option c is combinations. If someone thought that the order doesn't matter, but in reality, the order does matter because each day is a different day. So choosing cake A on day 1 and cake B on day 2 is different from the reverse. Therefore permutations are needed, not combinations. So option c is wrong because it uses combinations instead of permutations.

Therefore, the correct answer is e.

Alright! Let's tackle this problem step by step. The question is about ลลิษา buying desserts from a shop that sells two types: cakes with 8 kinds and pastries with 9 kinds. She buys one kind each day without repetition throughout the week. We need to find the total number of ways she can buy desserts in one week.

First, I need to understand the constraints. ลลิษา starts by choosing either a cake or a pastry on the first day. Once she picks a type, she must continue buying that same type every day for the entire week without repeating any kind. So, if she starts with a cake, she has to buy a different cake each day for seven days. Similarly, if she starts with a pastry, she must buy a different pastry each day for seven days.

Now, let's break it down into two scenarios:

1. Starting with Cake (8 kinds):
   Since there are 8 types of cakes and she needs to buy one each day for 7 days without repetition, we need to calculate the number of permutations of 8 cakes taken 7 at a time. The formula for permutations is P(n, k) = n! / (n - k)! where n is the total number of items, and k is the number of items to choose. Here, P(8, 7) = 8! / (8 - 7)! = 8! / 1! = 40320.

2. Starting with Pastry (9 kinds):
   Similarly, for pastries, there are 9 types, and she needs to buy one each day for 7 days without repetition. Using the permutation formula again, P(9, 7) = 9! / (9 - 7)! = 9! / 2! = 181440.

To find the total number of ways, we add the permutations from both scenarios:
Total ways = P(8, 7) + P(9, 7) = 40320 + 181440 = 221760.

Looking at the answer choices, option E is 8! + 9!/2. Let's compute that:
8! = 40320
9! = 362880
9!/2 = 181440
Adding them together: 40320 + 181440 = 221760, which matches our calculation.

Therefore, the correct answer is option E.