Chapter 11, Problem 13CR

To determine

To Find : The number of combinations of $5C2$ and $12C5$

Answer to Problem 13CR

The number of combinations of $5C2$ and $12C5$ are $10$ and $792$ respectively.

Explanation of Solution

Given information :

Consider the information provided in the question.

Formula used :

To find the number of combinations of $n$ objects taken $r$ at a time, divide the number of permutations of $n$ objects taken $r$ at a time by $r!$ can be written as :

  $nCr=\frac{n\mathrm{Pr}}{r!}$

Calculation :

Use the formula provided above to find $5C2$ and $12C5$

  $\begin{array}{l}nCr=\frac{n\mathrm{Pr}}{r!}\\ 5C2=\frac{5P2}{2!}\text{ [use combinations formula]}\\ \text{ =}\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{3\cdot 2\cdot 1\cdot 2\cdot 1}\text{ [write }5P2\text{ and }2!\text{ as products and divide out common factors ]}\\ \text{ = 5}\cdot \text{2}=10\text{ [multiply]}\end{array}$

  $\begin{array}{l}\\ nCr=\frac{n\mathrm{Pr}}{r!}\\ 12C5=\frac{12P5}{5!}\text{ [use combinations formula]}\\ \text{ =}\frac{12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}\text{ [write }5P2\text{ and }2!\text{ as products and divide out common factors ]}\\ \text{ = 11}\cdot \text{9}\cdot \text{8}=792\text{ [multiply]}\end{array}$

The number of combinations of $5C2$ and $12C5$ are $10$ and $792$ respectively.