I don't have a proof, but I know intuitively that it's true.Prove that: .
If there are identical red balls and identical blue balls,
. . in how many ways can they be arranged in a row?
The answer is: . . which is an integer.
As a last part of a solution of a second year algebra problem, I have to prove the identity:
a!b! | (a+b)!
a factorial times b factorial divides a+b factorial.
Anyone have any thoughts on how to tackle it? I tried 'writing out' the (a+b)! part, then cutting of either tail of a! or b!, but to no avail....
Thanks in advance~!
The product of n consecutive integers is divisible by n!
Consider the expansion of (a+b)!. It has (a+b) factors.
The product of the first a factors (or last a factors) is divisible by a!. The product of the last b factors (or first b factors) is divisible by b!.
Thus, (a+b)! is divisible by a!b!.
ie a!b! | (a+b)!