Final answer:
To buy a flower arrangement with exactly 4 types of flowers and 1 vase, there are 136,080 different arrangements. To buy a flower arrangement with exactly 11 types of flowers and 7 vases, there are 84 different arrangements. The number of different arrangements is smaller in part b compared to part a.
Step-by-step explanation:
a. To determine the number of different arrangements, we can use the concept of combinations. We have 4 types of flowers to choose from and we need exactly 1 vase. So, we can choose 4 flowers out of 15, which can be done in C(15,4) ways. Additionally, we can choose 1 vase out of 8 in C(8,1) ways. To find the total number of different arrangements, we multiply these two combinations together. Therefore, the number of different arrangements you can buy is C(15,4) * C(8,1) = 136,080.
b. Similarly, for part b, we have 11 types of flowers and we need 7 vases. So, we can choose 11 flowers out of 15 in C(15,11) ways. Additionally, we can choose 7 vases out of 8 in C(8,7) ways. Multiplying these two combinations together gives us the total number of different arrangements, which is C(15,11) * C(8,7) = 84.
c. When comparing the solutions to parts (a) and (b), we can see that the number of different arrangements in part (b) is much smaller than in part (a). This is because we are choosing fewer flowers and more vases in part (b), which limits the number of possible combinations.