1. ## Binomial Coefficients Formula

Proof that ${n\choose{k}} = \frac{n!}{k!(n-k)!} = \frac{(n-k+1)(n-k+2)\cdot\cdot\cdot(n-1)n}{2\cdot{3}\cdot\cdot\cdot(k-1)k}$

2. Originally Posted by Mozart
Proof that ${n\choose{k}} = \frac{n!}{k!(n-k)!} = \frac{(n-k+1)(n-k+2)\cdot\cdot\cdot(n-1)n}{2\cdot{3}\cdot\cdot\cdot(k-1)k}$
You can write the $n!$ in the numerator as

$2\cdot 3 \cdot 4 \cdot \dots \cdot (n - k)\cdot(n - k + 1)\cdot(n - k + 2)\cdot \dots (n - 1)\cdot n$

The $(n - k)!$ in the denominator is

$2 \cdot 3 \cdot 4 \cdot \dots \cdot (n - k)$.

Can you see anything that cancels?

Also, $k! = 2 \cdot 3 \cdot 4 \cdot \dots \cdot k$.