Question

Evaluate$\left(\begin{array}{c}37\\ 19\end{array}\right)$

Short answer

The value of$\left(\begin{array}{c}37\\ 19\end{array}\right)=17672631900$

Step-by-step solution

Step 1. Defination

if $j$and $n$are the 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

$\Rightarrow \left(\begin{array}{c}37\\ 19\end{array}\right)=\frac{37!}{19!\left(37-19\right)!}$

$\Rightarrow \left(\begin{array}{c}37\\ 19\end{array}\right)=\frac{37\times 36\times 35\cdots \cdots \times 1}{\left(19\times 18\times 17\cdots \cdots \times 1\right)\left(18\times 17\times 16\cdots \cdots \times 1\right)}$

$\Rightarrow \left(\begin{array}{c}37\\ 19\end{array}\right)=17672631900$