Question

Evaluate$\left(\begin{array}{c}1000\\ 0\end{array}\right)$

Short answer

The value of$\left(\begin{array}{c}1000\\ 0\end{array}\right)=1$

Step-by-step solution

Step 1. Defination

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

$\Rightarrow \left(\begin{array}{c}1000\\ 0\end{array}\right)=\frac{1000!}{0!\left(1000-0\right)!}$

localid="1647003613868" $\Rightarrow \left(\begin{array}{c}1000\\ 0\end{array}\right)=\frac{1000\times 999\times 998\cdots \cdots \times 1}{\left(0!\right)\left(1000\times 999\times 998\cdots \cdots \times 1\right)}$

localid="1647003620873" $\Rightarrow \left(\begin{array}{c}1000\\ 0\end{array}\right)=1$