Chapter 12, Problem 27CR

To determine

Find the number of permutations.

Answer to Problem 27CR

  $42$

Explanation of Solution

Given information:

  ${}_7{P}_2$

Calculation:

Consider

  ${}_7{P}_2$

Use the definition of permutations.

Permutation is defined as, the arrangement of object in a specific order. For any positive integer’s $n,\&r$ where $r\le n,$ the arrangement of $n$ objects taking $r$ at time is denoted by ${}_n{P}_r$ and defined by;

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

Put the value, $7$ for $n$ , and $2$ for $r$ , in the equation ${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

  $=\frac{7!}{5!}$

  ${}_7{P}_2=\frac{7!}{\left(7-2\right)!}$ use the definition of $n$ factorial $n!=n\cdot \left(n-1\right)\cdot \mathrm{...}\cdot 3\cdot 2\cdot 1=\frac{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{5\cdot 4\cdot 3\cdot 2\cdot 1}$

  $=42$

Hence, the expression ${}_7{P}_2$ equals to $42$ .