Chapter 12.6, Problem 56PPE

To determine

To calculate: the value of expression.

Answer to Problem 56PPE

The value is $3$ .

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 ,

  $\frac{P{}_3}{P{}_2}$

  $\begin{array}{l}P{}_3=\frac{5!}{\left(5-3\right)!}\\ =\frac{5!}{2!}\\ =\frac{5\times 4\times 3\times 2\times 1}{2\times 1}\\ =5\times 4\times 3\\ =60\end{array}$

  $\begin{array}{l}P{}_2=\frac{5!}{\left(5-2\right)!}\\ =\frac{5!}{3!}\\ =\frac{5\times 4\times 3\times 2\times 1}{3\times 2\times 1}\\ =5\times 4\\ =20\end{array}$

Now,

  $\begin{array}{l}\frac{P{}_3}{P{}_2}=\frac{60}{20}\\ =3\end{array}$

Hence, the value is $3.$