Solved: Proving Distributions of Musicians w/ Induction

[rsa58]

[SOLVED] induction? question

Homework Statement

the question is:
let k, n, and k1, . . . , kn be given natural numbers, such that
k1 + . . . + kn = k.
Assume that k musicians shall be distributed to n orchestras such that exactly
ki musicians play in the ith orchestra. Prove that there exist exactly
k!/(k1! · · · kn!)
different distributions.

is it possible to use induction to answer this? i can prove it by using the choose function to find all the possible distributions. in that way i get a proof for the statement, but i am unable to assume that it is correct for n, and then show it is correct for n+1. can someone give me some ideas?

Homework Equations

The Attempt at a Solution

 

[EnumaElish]

in that way i get a proof for the statement

Why do you need induction if you have a proof without it?

 

[rsa58]

because the professor said something about induction. he said to "select from a pool" (of k's i guess, whatever that means) and then use induction over n. he said to check for n=1 and n=2 and show that the statement holds for n>= 2. he does seem kind of out of it though. to use induction how would i proceed? i need to combine some expression with
k!/(k1! · · · kn!)

right? i can't find this expression. the only sort of induction i can get is a logically inherent one using the choose function.