Question

Derek is planning a banquet for 150150 guests. He needs to select 55 appetizers, 44 entrees, and 44 side dishes. Carson's Catering has 1616 appetizers, 88 entrees, and 77 side dishes to choose from. How many different ways are there for Derek to plan the menu?

Answer 1

There are 10,701,600 different ways for Derek to plan the menu.

Explanation:

We use the combinations formula, because the order that the appetizers, entrees and side dishes are going to be selected does not matter. For example, appetizers A then appetizer B is the same outcomes as appetizer B then appetizer A.

Combinations formula:

`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

Derek will choose

5 appetizers from a set of 16.

4 entrees from a set of 8.

4 side dishes from a set of 7.

How many different ways are there for Derek to plan the menu?

We multiply the number of possible outcomes for appetizers, entrees and side dishes.

`T = C_(16,5)*C_(8,4)*C_(7,4) = (16!)/(11!5!)*(8!)/(4!4!)*(7!)/(3!4!) = 4368*70*35 = 10701600`

There are 10,701,600 different ways for Derek to plan the menu.