Question

How do you evaluate ${}^{10}C_3$?

Answer 1

Evaluating the expression ${}^{10}C_3$ :

As we know, a combination is a grouping or subset of items.

We will evaluate the given expression ${}^{10}C_3$ by using the combination formula.

The order does not matter in a combination.

So,

$\Rightarrow$ $C\left(n,r\right)={}^{n}C_r$

$\Rightarrow$ ${}^{n}C_r=\frac{n!}{\left(n-r\right)!r!}$

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

$\Rightarrow$ ${}^{10}C_3=\frac{10!}{\left(10-3\right)!\times \left(3!\right)}$

$\Rightarrow$ $\frac{10\times 9\times 8\times 7\times 6\times ...\times 1}{7\times 6\times ...\times 1·\left(3\times 2\times 1\right)}$

$\Rightarrow$ $\frac{10\times 9\times 8}{3\times 2\times 1}$

$\Rightarrow$ ${}^{10}C_3=120$

Hence, the value of ${}^{10}C_3$ is $120$.