guide to factorials

simplifying factorials.

one thing to notice: $n!=(n)(n-1)!$

this is how you break down a factorial. we usually break down the larger factorial

worked example: simplify $\frac{8!}{6!}$
$\frac{(8)(7)(6!)}{6!}$
$56$

worked example: simplify $\frac{3!7!}{6!4!0!}$
$\frac{(3!)(7)(6!)}{(6!)(4)(3!)(1)}$
$\frac{7}{4}$

worked example: simplify $\frac{(3n+2)!}{(3n-1)!}$
$\frac{(3n+2)(3n+1)(3n)(3n-1)!}{(3n-1)!}$
$(3n+2)(3n+1)(3n)$

worked example: simplify $\frac{(n+2)!-(n)!}{(n+1)!}$
$\frac{(n+2)!}{(n+1)!}-\frac{n!}{(n+1)!}$
$\frac{(n+2)(n+1)!}{(n+1)!}-\frac{n!}{(n+1)(n)!}$
$n+2-\frac{1}{n-1}$