Question

What is ${}^5{C}_3$?

Answer 1

Determine ${}^5{C}_3$.

we know,${}^n{C}_r=\frac{n!}{r!(n-r)!}$

To calculate, put $n=5,r=3$ in the above formula:

$$\begin{gathered} {}^n{C}_r=\frac{n!}{r!(n-r)!} \\ \Rightarrow {}^5{C}_3=\frac{5!}{3!(5-3)!} \\ \Rightarrow {}^5{C}_3=\frac{5!}{3!(5-3)!} \\ \Rightarrow {}^5{C}_3=\frac{5!}{3!\left(2\right)!} \\ \Rightarrow {}^5{C}_3=\frac{5\times 4\times 3!}{3!\times 2\times 1} \\ \Rightarrow {}^5{C}_3=\frac{5\times 4}{2\times 1} \\ \Rightarrow {}^5{C}_3=5\times 2 \\ \Rightarrow {}^5{C}_3=10 \end{gathered}$$

Hence, the required value is ${}^5{C}_3=10$