Question

How do you evaluate ${}^{5}C_2+{}^{5}C_1$?

Answer 1

Estimate the value of the expression:

We will evaluate ${}^{5}C_2+{}^{5}C_1$ by using the combination formula.

As we know, a combination is a grouping or subset of items. The order doesn't matter here.

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

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

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

$\Rightarrow$ ${}^{5}C_2+{}^{5}C_1=\frac{5!}{\left(5-2\right)!2!}+\frac{5!}{\left(5-1\right)!1!}$

$\Rightarrow$ ${}^{5}C_2+{}^{5}C_1=\frac{5\times 4\times 3!}{3!2!}+\frac{5\times 4!}{4!1!}$

Cancel the common factors and simplify them.

$\Rightarrow$ ${}^{5}C_2+{}^{5}C_1=\frac{5\times 4\times \cancel{3}!}{\cancel{3}!2!}+\frac{5\times \cancel{4}!}{\cancel{4}!1!}$

$\Rightarrow$ ${}^{5}C_2+{}^{5}C_1=\frac{5\times 4}{2!}+\frac{5}{1!}$

$\Rightarrow$ ${}^{5}C_2+{}^{5}C_1=\frac{5\times 4}{2}+\frac{5}{1}$

$\Rightarrow$ ${}^{5}C_2+{}^{5}C_1=10+5$

$\Rightarrow$ ${}^{5}C_2+{}^{5}C_1=15$

Hence, the value for the equation ${}^{5}C_2+{}^{5}C_1$ is $15$.