Chapter 12.6, Problem 41PPE

To determine

To show: which value is greater.

Answer to Problem 41PPE

  $P{}_7\text{ is greater than }P{}_2.$

Explanation of Solution

Given information:

  $P{}_7\text{ or }P{}_2$

Formula used:

A permutation is an arrangement of objects in a specific order. The expression $P{}_r$ represents the number of permutations of $n$ objects arranged $r$ at a time.

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

For any positive integer $n$, the expression $n$ factorial is written as $n!$ and is the product of integers from n down to 1.

Calculation:

Consider ,

  $P{}_7\text{ or }P{}_2$

  $\begin{array}{l}P{}_7=\frac{9!}{\left(9-7\right)!}\\ =\frac{9!}{2!}\\ =\frac{9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{2\times 1}\\ =9\times 4\times 7\times 6\times 5\times 4\times 3\times 2\times 1\\ =181,440\end{array}$

  $\begin{array}{l}P{}_2=\frac{9!}{\left(9-2\right)!}\\ =\frac{9!}{7!}\\ =\frac{9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{7\times 6\times 5\times 4\times 3\times 2\times 1}\\ =9\times 8\\ =72\end{array}$

Hence, $P{}_7\text{ is greater than }P{}_2.$ .