simplify factorials: $\frac{(k-1)!}{(k+2)!}$

Question

Question: simplify $$\frac{(k-1)!}{(k+2)!}$$

What I did was:

$$\frac{(k - 1)!k!}{(k + 2)! \cdot (k + 1)!}$$ This I did following the rule $n! = n \times (n - 1)!$.

can this be simplified further? Thanks.

Answer 1

One may write $$ \frac{(k-1)!}{(k+2)!}=\frac{1\cdot\color{blue}{(k-1)!}}{(k+2)(k+1)k\:\color{blue}{(k-1)!}}=\frac1{(k+2)(k+1)k}. $$

Answer 2

HInt: $(k+2)!=(k+2)(k+1)k(k-1)!$