Question

Evaluate$\left(\begin{array}{c}5\\ 3\end{array}\right)$

Short answer

The value of$\left(\begin{array}{c}5\\ 3\end{array}\right)=10$.

Step-by-step solution

Step 1. Definition

If $j$and $n$are integers with $0\le j\le n$, the symbol$\left(\begin{array}{c}n\\ j\end{array}\right)$is defined as$\left(\begin{array}{c}n\\ j\end{array}\right)=\frac{n!}{j!\left(n-j\right)!}$.

Step 2. Explanation

$\Rightarrow \left(\begin{array}{c}5\\ 3\end{array}\right)=\frac{5!}{3!\left(5-3\right)!}$

$\Rightarrow \left(\begin{array}{c}5\\ 3\end{array}\right)=\frac{5\times 4\times 3\times 2\times 1}{\left(3\times 2\times 1\right)\left(2\times 1\right)}$

$\Rightarrow \left(\begin{array}{c}5\\ 3\end{array}\right)=10$