Chapter 12, Problem 32CR

To determine

Find the number of combinations.

Answer to Problem 32CR

  ${}_7{C}_3=35$

Explanation of Solution

Given information:

  ${}_7{C}_3$

Calculation:

Consider

  ${}_7{C}_3\mathrm{......}\left(1\right)$

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;

  ${}_n{C}_r=\frac{n!}{r!(n-r)!}\mathrm{......}(2)$

Put the value, $7$ for $n$ , and $3$ for $r$ , in the equation $(2)$ .

  ${}_7{C}_3=\frac{7!}{3!\left(7-3\right)!}$

  $=\frac{7!}{3!4!}$ The $n$ factorial is defined as: $n!=n\cdot (n-1)\cdot \mathrm{...}\cdot 3\cdot 2\cdot 1$

  $\begin{array}{l}=\frac{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{\left(3\cdot 2\cdot 1\right)\left(4\cdot 3\cdot 2\cdot 1\right)}\\ =35\end{array}$

Hence, the number of combinations of the expression ${}_7{C}_3$ is ${}_7{C}_3=35$ .