Chapter 8.5, Problem 44E

To determine

To find: the number of different orders of the top three finishes are possible.

Answer to Problem 44E

The number of permutation of six sprinters taken three at a time is

  ${}_8{P}_3=336$

Explanation of Solution

Given information:

8 sprinters have qualified for the finals in the 100 meter dash at the NCAA national track meet.

Calculation:

Assume there are 8 sprinters qualified for final round in a competition, also assume that only 3 prices are given, and so find the number of ways of selection of first 3 among 8, since it is a arrangement of 3 from 8 people, therefore,

  ${}_8{P}_3$

Since the number of permutations of $n$ elements taken $r$ at a time is,

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

Here $n=8,r=3$ , hence put the values in the formula and solve as,

  $\begin{array}{c}{}_8{P}_3=\frac{8!}{\left(8-3\right)!}\\ {}_8{P}_3=\frac{8!}{5!}\\ =\frac{8\cdot 7\cdot 6\cdot 5!}{5!}\\ =336\end{array}$

So, the number of permutation of six sprinters taken three at a time is

  ${}_8{P}_3=336$