A caterer offers a choice of five desserts from a list of eight for the dessert bar at the prom. How many different groups of desserts could be served?
You're choosing 5 desserts from 8, so it is (8 choose 5).
6720
To find the number of different groups of desserts that could be served, we can use the concept of combinations.
In this case, we need to select a group of desserts from a list of eight, and we can choose any number from one to five, as the caterer offers a choice of five desserts.
The number of different groups of desserts can be calculated using the formula for combinations:
C(n, r) = n! / (r!(n - r)!)
In this formula, n represents the total number of items to choose from (in this case, eight desserts), and r represents the number of items to select (in this case, the number of desserts to be served, ranging from one to five).
Let's calculate the different groups of desserts:
For r = 1:
C(8, 1) = 8! / (1! * (8 - 1)!) = 8! / (1! * 7!) = 8
For r = 2:
C(8, 2) = 8! / (2! * (8 - 2)!) = 8! / (2! * 6!) = (8 * 7) / (2 * 1) = 28
For r = 3:
C(8, 3) = 8! / (3! * (8 - 3)!) = 8! / (3! * 5!) = (8 * 7 * 6) / (3 * 2 * 1) = 56
For r = 4:
C(8, 4) = 8! / (4! * (8 - 4)!) = 8! / (4! * 4!) = (8 * 7 * 6 * 5) / (4 * 3 * 2 * 1) = 70
For r = 5:
C(8, 5) = 8! / (5! * (8 - 5)!) = 8! / (5! * 3!) = (8 * 7 * 6 * 5 * 4) / (5 * 4 * 3 * 2 * 1) = 56
So, the caterer could serve a total of 8 + 28 + 56 + 70 + 56 = 218 different groups of desserts.