Question

evaluate the following combination: $_{7}c_{4}$

Explanation:

Step1: Recall combination formula

The combination formula is $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 7$ and $r=4$.

Step2: Calculate factorial values

$n!=7! = 7\times6\times5\times4\times3\times2\times1$, $r!=4!=4\times3\times2\times1$, and $(n - r)!=(7 - 4)!=3!=3\times2\times1$. Then $C(7,4)=\frac{7!}{4!(7 - 4)!}=\frac{7!}{4!3!}=\frac{7\times6\times5\times4!}{4!\times3\times2\times1}$.

Step3: Simplify the expression

Cancel out the $4!$ terms. We get $\frac{7\times6\times5}{3\times2\times1}=\frac{210}{6}=35$.

Answer:

35