Chapter 9.6, Problem 42E

To determine

To find: the number of ways can the sprinters come in first, second and third.

Answer to Problem 42E

The number of permutation of six sprinters taken three at a time is ${}_6{P}_3=120$ .

Explanation of Solution

Given information:

Eight sprinters qualify for the finals in the 100 meter dash at the NCAA national track meet.

Calculation:

Assume there are 6 sprinters qualified for final round in a competition, also assume that only 3 price are given, and so find the number of ways of selection of first 3 among 6, since it is a arrangement of 3 from 6 people,

  ${}_6{P}_3$

The number of permutations of $n$ elements taken $r$ at a time is

  ${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

Here $n=6,r=3$ , hence put the values in the formula and solve as,

  $\begin{array}{c}{}_6{P}_6=\frac{6!}{\left(6-3\right)!}\\ =\frac{6!}{3!}\\ =\frac{6\times 5\times 4\times 3}{3!}\\ =120\end{array}$

Hence, the number of permutation of six sprinters taken three at a time is ${}_6{P}_3=120$ .