Chapter 8, Problem 70CR

To determine

To solve: The value of $n$ in $15·P{}_2=P{}_3$ .

Answer to Problem 70CR

The value of $n$ is $14$ .

Explanation of Solution

Given:

The equation is $15·P{}_2=P{}_3$ .

Formula used:

The number of permutation of $n$ elements taken $r$ at a time is,

  $\begin{array}{c}P{}_r=\frac{n!}{\left(n-r\right)!}\\ =n\left(n-1\right)\left(n-2\right)\cdots \left(n-r+1\right)\end{array}$

Calculation:

Consider the equation,

  $15·P{}_2=P{}_3$

From permutations, the equation is rewritten as follows:

  $15·\frac{n!}{\left(n-2\right)!}=\frac{\left(n+1\right)!}{\left(n+1-3\right)!}$

Calculate the value of $n$ as follows:

  $\begin{array}{c}15·\frac{n!}{\left(n-2\right)!}=\frac{\left(n+1\right)!}{\left(n-2\right)!}\\ 15n!=\left(n+1\right)n!\\ n+1=15\\ n=14\end{array}$

Therefore, the value of $n$ is $14$ .