Chapter 6.1, Problem 40E

Solution

To find the number of ways the coach can select 4 students out of 20 to compete on the relay team.

The number of ways the coach can select four students to compete the race is $\text{116280}$.

Given information:

Four students would compete on different parts of the race. There are total 20 students.

Formula used:

  $C{}_r=\frac{n!}{\left(n-r\right)!\cdot r!}$

Calculation:

Four students out of 20 students can be chosen in $C{}_5$ ways.

These four students can be placed at different parts of the race in $4!$ ways.

Therefore, the total number of ways the coach can select four students to compete the race is $C{}_5\times 4!$.

Evaluate $C{}_5\times 4!$.

  $\begin{array}{c}C{}_5\times 4!=\frac{20!}{\left(20-4\right)!\cdot 4!}\times 4!\\ =\frac{20!}{16!\cdot 4!}\times 4!\\ =\frac{20\cdot 19\cdot 18\cdot 17\cdot 16!}{16!\cdot 4!}\times 4!\\ =\frac{20\cdot 19\cdot 18\cdot 17}{4!}\times 4!\\ =20\cdot 19\cdot 18\cdot 17\\ =\text{116280}\end{array}$

Conclusion:

The number of ways the coach can select four students to compete the race is $\text{116280}$.