Question

Find the number of subsets of the set $1,2,3,4,5,6,7,8,9,10,11$ having $4$ elements.

Answer 1

Calculate the number of subsets of the set

A combination is a grouping of outcomes in which the order does not matter.

The number of combinations of $\mathrm{n}$ things chosen $\mathrm{r}$ at a time is found by using the following formula,

${}^{\mathbf{n}}\mathbf{C}_{\mathbf{r}}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{r}\mathbf{!}\left(n-r\right)\mathbf{!}}$

Here, $\mathrm{n}$ number of items in set and $\mathrm{r}$ is the number of items selected from the set.

Given Number of digits $=11$

The sequence is, $1,2,3,4,5,6,7,8,9,10,11$

Let,

$\mathrm{S}=\left\{1,2,3,4,5,6,7,8,9,10,11\right\}$

The number of subsets of $\mathrm{S}$ containing exactly $4$ elements which means the number of ways we can select $4$ element from $11$ elements are,

${}^{11}C_4=\frac{11!}{4!\times \left(11-4\right)!}$

$=\frac{11!}{4!\times 7!}$

$=\frac{11\times 10\times 9\times 8}{4\times 3\times 2}$

$\therefore$ ${}^{11}C_4=330$

Hence, the number of subsets are $330$.