Chapter 11.7, Problem 7E

To determine

Whether the number of teams the coach can select can be counted using permutations or combinations and then find the number of teams.

Answer to Problem 7E

The number of teams that can be formed are $1001$ .

Explanation of Solution

Given information:

A gym coach selects a team of $4$ athletes from $14$ athletes.

The order in which the members of a team are selected does not matter. So, the possibilities can be counted using combination. So to find the number of different ways to select a team of $4$ athletes from $14$ , find $14C4$ .

Calculation:

To find the number of combinations of $n$ objects taken $r$ at a time, divide the number of permutations of $n$ objects taken $r$ at a time by $r!$

  $nCr=\frac{nPr}{r!}$   [Combination formula]

  $\Rightarrow 14C4=\frac{14P4}{4!}$   [Use combination formula]

  $=\frac{14\cdot 13\cdot 12\cdot 11}{4\cdot 3\cdot 2\cdot 1}$   [Write $14P4$ and $4!$ as a product]

  $=\frac{\cancel{\stackrel{7}{14}}\cdot 13\cdot \cancel{\stackrel{\cancel{2}\cancel{6}}{12}}\cdot 11}{\cancel{\underset{\cancel{2}}{4}}\cdot \cancel{\underset{1}{3}}\cdot \cancel{\underset{1}{2}}\cdot 1}$   [Divide out common factors]

  $=1001$   [Simplify]

Therefore, the number of combinations are $1001$ .