Chapter 12.6, Problem 57PPE

To determine

To calculate: the value of expression.

Answer to Problem 57PPE

The value is $2$ .

Explanation of Solution

Given information:

  $\frac{P{}_3}{P{}_2}$

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 ,

  $\frac{P{}_3}{P{}_2}$

  $\begin{array}{l}P{}_3=\frac{4!}{\left(4-3\right)!}\\ =\frac{4!}{1!}\\ =\frac{4\times 3\times 2\times 1}{1}\\ =4\times 3\times 2\\ =24\end{array}$

  $\begin{array}{l}P{}_2=\frac{4!}{\left(4-2\right)!}\\ =\frac{4!}{2!}\\ =\frac{4\times 3\times 2\times 1}{2\times 1}\\ =4\times 3\\ =12\end{array}$

Now,

  $\begin{array}{l}\frac{P{}_3}{P{}_2}=\frac{24}{12}\\ =2\end{array}$

Hence, the value is $2.$