Chapter 12, Problem 30CR

To determine

Find the number of combinations.

Answer to Problem 30CR

  $10$

Explanation of Solution

Given information:

  ${}_5{C}_3$

Calculation:

Consider

  ${}_5{C}_3$

Use the definition of combination.

Combination is defined as, the selection or combination of objects without any order. For any positive integer’s $n,and,r,$ the selection of $n$ objects taking $r$ at time, where $r\le n,$ is denoted by ${}_n{C}_r$ and defined by;

  ${}_n{C}_r=\frac{n!}{r!(n-r)!}$

Put the value, $5$ for $n$ , and $3$ for $r$ , in the equation.

  ${}_5{C}_3=\frac{5!}{3!\left(5-3\right)!}$

  $=\frac{5!}{3!2!}$ The $n$ factorial is defined as: $n!=n\cdot (n-1)\cdot \mathrm{...}\cdot 3\cdot 2\cdot 1$

  $\begin{array}{l}=\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{\left(3\cdot 2\cdot 1\right)\left(2\cdot 1\right)}\\ =10\end{array}$

Hence, the number of combinations of the expression ${}_5{C}_3$ is $10$ .