Chapter 12.6, Problem 42PPE

To determine

To show: which value is greater.

Answer to Problem 42PPE

  $P{}_4\text{ is greater than }P{}_3.$

Explanation of Solution

Given information:

  $P{}_3\text{ or }P{}_4$

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 ,

  $P{}_3\text{ or }P{}_4$

  $\begin{array}{l}P{}_3=\frac{10!}{\left(10-3\right)!}\\ =\frac{10!}{7!}\\ =\frac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{7\times 6\times 5\times 4\times 3\times 2\times 1}\\ =10\times 9\times 8\\ =720\end{array}$

  $\begin{array}{l}P{}_4=\frac{8!}{\left(8-4\right)!}\\ =\frac{8!}{4!}\\ =\frac{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{4\times 3\times 2\times 1}\\ =8\times 7\times 6\times 5\\ =1,680\end{array}$

Hence, $P{}_4\text{ is greater than }P{}_3.$ .