Question

What is ${}^{6}C_4$ combination?

Answer 1

Combination:

${}^{6}C_4$ means choose(selection) $4$ elements from a set of $6$ elements.

The formula of combination: ${}^{\mathbf{n}}\mathbf{C}_{\mathbf{r}}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{r}\mathbf{!}\left(n-r\right)\mathbf{!}}$

Therefore,

${}^{6}C_4=\frac{6!}{4!\left(6-4\right)!}$

$\Rightarrow$${}^{6}C_4=\frac{6!}{4!\left(2\right)!}$

$\Rightarrow$${}^{6}C_4=\frac{6\times 5\times 4!}{4!\left(2\right)!}$

$\Rightarrow$${}^{6}C_4=\frac{30}{2}$

$\Rightarrow$ ${}^{6}C_4=15$

Hence, ${}^{6}C_4$ means there are$15$ ways to select 4 objects from 6 different objects.