Number 30. Zeta wants to buy some flowers. The prices of the three flowers she can buy are shown in the picture. How many different bouquets with a total cost of exactly 23 can she buy? Pause if you would like to try and solve it. Alright, so the key here is to just count the different options we have, all the different bouquets she can buy, and try to make sure we don't miss any. So for that, we can use a table to keep track of all of the different bouquets that we come up with. And this will help us make sure that we don't miss any. Alright, so we'll have the number of $3 flowers, the number of $4 flowers, and the number of $5 flowers. Alright, so what's the largest amount of $3 flowers she can buy? Well, she can't buy 8, because 8 times 3 is 24, which is more than 23. She can't buy 7, because 3 times 7 is 21, which only leaves her $2 left over, and there's no $2 flowers here. But she can buy 6. 6 times 3 is 18. 23 minus 18 is 5, which means she has enough money to buy exactly one $5 flower. And then, of course, no $4 flowers. Alright, so what if she now buys 5 $3 flowers? 5 times 3 is 15. 23 minus 15 is 8, which is enough for two $4 flowers, and no $5 flowers. And then we can also quickly think there's no other options for her to make with exactly $8 left over. Alright, so what if she gets 4 $3 flowers? Well, 4 times 3 is 12. 23 minus 12 is 11. And we can actually see that if she only has exactly $11, there's no combination of 4 and $5 flowers that will add up to 11, right? If she gets two $5 flowers, that's 10, which means she won't be able to get to 11. If she gets two $4 flowers, that's 8. That also doesn't get there. One $4, one $5, that's 9. She's not going to be able to get exactly $11 worth of flowers, which means that actually 4 is not possible. She can't buy exactly four $3 flowers. She has to buy either more or less. Alright, so what if she buys less, because we've already checked more? If she gets three $3 flowers, 3 times 3 is 9. 23 minus 9 is 14, which is exactly enough for one $4 flower and two $5 flowers, right? One $4 flower is 4. 2 times 5 is 10. 4 plus 10 is 14. And that's the remaining 14 that's left over after the three $3 flowers, which are worth 9. Alright, so what if she buys then two $3 flowers? Well, 2 times 3 is 6. 23 minus 6 is 17. And so with 17, this might take a minute to think about it, but you can get that adding up only 4 and 5. And you can get that by adding up three $4 flowers with one $5 flower, right? 3 times 4 is 12. 5 times 1 is 5. 12 plus 5 is 17. And of course, 17 plus 2 times 3, which is 6. 17 plus 6 is 23. Alright, so what if she buys one $3 flower? Well, 1 times 3 is 3. 23 minus 3 is 20. And there's actually two ways that she can get 20 using only 4 and $5 flowers. She can either do five $4 flowers and no $5 flowers. Or she can do no $4 flowers and four $5 flowers, right? Because either way, 4 times 5 and 5 times 4, they're both 20. And 20 plus the remaining one $3 flower is 23. Alright, so there's one last case. And that is if she buys no $3 flowers. Are there any ways to add up to 23 using just 4 and $5 flowers? And there actually is one way. And it's if she gets two $4 flowers and three $5 flowers. 2 times 4 is 8. 5 times 3 is 15. 15 plus 8 is 23. So these are all the different ways. We've checked every amount of $3 flowers she can get. And since there's no other ways, we know this is all of the possibilities. And so we can just count how many there are. There are 1, 2, 3, 4, 5, 6, 7. Which means the answer to number 30 is option D, 7.

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Zeta

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