Question

evaluate the following combination: $_{9}c_{7}$

Explanation:

Step1: Recall combination formula

The combination formula is $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 9$ and $r=7$.

Step2: Calculate $n - r$

$n-r=9 - 7=2$.

Step3: Substitute values into formula

$_{9}C_{7}=\frac{9!}{7!(9 - 7)!}=\frac{9!}{7!2!}$.
Since $n!=n\times(n - 1)\times\cdots\times1$, $9! = 9\times8\times7!$ and $2! = 2\times1$.
So $\frac{9!}{7!2!}=\frac{9\times8\times7!}{7!\times2\times1}$.

Step4: Simplify the expression

Cancel out the $7!$ terms in the numerator and denominator. We get $\frac{9\times8}{2\times1}=\frac{72}{2}=36$.

Answer:

36