Number of ways to write a number $N$ as the sum of $M$ natural numbers, where order doesn't matter?

Question

Is there a formula to write a number $N$ as the sum of $M$ natural numbers (not necessarily distinct), where order DOES NOT matter? I know that you can use ${N-1}\choose{M-1}$ to find the number of ways counting order, but I'm looking for a formula that defines, for example, $1+1+2$ and $1+2+1$ as the same.

1. How many ways are there to write 13 as the sum of 4 natural numbers if order does not matter?

2. How many ways are there to write 8 as the sum of 4 natural numbers if order does not matter?

3. How many ways are there to write 11 as the sum of 5 natural numbers if order does not matter?

Thanks!

Answer 1

This is a question about partitions. One is asking how many partitions of $N$ are there into exactly $M$ parts. Call this $P(N,M)$ One answer is that there is a generating function $$\sum_{N=M}^\infty P(N,M)x^N=\frac{x^M}{(1-x)(1-x^2)\cdots(1-x^M)}$$ for each $M$.