Simplify $\frac{a!}{(a+b)!}$

Question

Is there a way to simplfy $\frac{a!}{(a+b)!}$?

What about $\frac{a!b!}{(a+b)!}$?

I have tried taking ${a}$ out of the top and bottom, getting $\frac{(a-1)!}{(a+b-1)!}$ but I can't reduce it down to remove $a$ completely.

Answer 1

You can write it explicitly and then simplify as follows : $$\frac{a!}{(a+b)!}=\frac{1}{(a+1)(a+2)\ldots (a+b)}$$

The same works for $\frac{a!b!}{(a+b)!}$ .

Also (maybe it helps you) :

$$\frac{a!b!}{(a+b)!}=\frac{1}{\binom{a+b}{a}}$$ where :

$\binom{n}{m}$ is a binomial coefficient and represents the numbers of ways $m$ objects can be chosen from $n$ objects and is generally given by the formula :

$$\binom{n}{m}=\frac{n!}{m!(n-m)!}$$

I don't think there's more to simplifying these expressions than this .

Maybe you can show us a problem where you tried to use this simplification . This way we can all better know how to help you .