Question

evaluate the following combination:
$_{11}c_{5}$

Explanation:

Step1: Recall combination formula

The combination formula is $_{n}C_{r}=\frac{n!}{r!(n - r)!}$, where $n = 11$ and $r=5$.

Step2: Calculate factorial values

$n!=n\times(n - 1)\times\cdots\times1$. So, $11! = 11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1$, $5! = 5\times4\times3\times2\times1$, and $(11 - 5)!=6!=6\times5\times4\times3\times2\times1$. Then $_{11}C_{5}=\frac{11!}{5!(11 - 5)!}=\frac{11!}{5!6!}=\frac{11\times10\times9\times8\times7\times6!}{5\times4\times3\times2\times1\times6!}$.

Step3: Simplify the expression

Cancel out the $6!$ terms. Then $\frac{11\times10\times9\times8\times7}{5\times4\times3\times2\times1}=\frac{11\times10\times9\times8\times7}{120}=462$.

Answer:

$462$