Chapter 13.1, Problem 41E

To determine

To calculate: The number of different fruit salads using 5 different fruits.

Answer to Problem 41E

The number of different fruit salads using 5 different fruitsare $2002$ .

Explanation of Solution

Given information:

There are 14 different varieties of fruit available to choose.

Formula used:

If there are n objects taken r at a time then combination is defined as $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$ .

Calculation:

Consider the provided information that there are 14 different varieties of fruit available to choose.

In order to make different fruit salads with 5 different fruits, use combination as selection does not matter.

Recall that if there are n objects taken r at a time then combination is defined as $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$ .

Here n is 14 and r is 5for $C\left(14,5\right)$ .

  $\begin{array}{c}C\left(14,5\right)=\frac{14!}{5!\left(14-5\right)!}\\ =\frac{14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9!}{5!\cdot 9!}\\ =\frac{14\cdot 13\cdot 12\cdot 11\cdot 10}{5\cdot 4\cdot 3\cdot 2\cdot 1}\\ =2002\end{array}$

Thus, the number of different fruit salads using 5 different fruits are $2002$ .