Given $m$ balls and $n$ baskets, in how many ways can we distribute the $m$ balls to the $n$ baskets when given that $n > m$ (the number of baskets is greater than the number of balls)?
So I said, that it does not differ if $n>m$ the solution is still ${n+m-1 \choose n-1}$, is that correct? I merely considered it as the number of solutions $x_1 + x_2 + … + x_n = m$ when $0 \leq x_i$. Is that correct?
Also, as an addition:
In addition to the question, what if there can be only a maximum of one ball in each basket?
So I said, it is still the same $x_1 + x_2 + … + x_n = m$ but this time $0 \leq x_i \leq 1$ and it is still the same solution. Is that correct?
Best Answer
We assume the usual convention: the balls are indistinguishable, and the baskets are distinguishable.
For your first question, yes, it is precisely the formula you are familiar with.
When we are restricted to at most one ball per basket, then yes, we are counting the solutions of $x_1+x_2+\cdots+x_n=m$ with the restriction that for any $i$ we have $x_i=0$ or $x_i=1$.
But to me the easiest way to think of it is that we are choosing the $m$ baskets that will be lucky and get a ball. Or if you wish choosing the $i$ such that $x_i=1$. Or else think of it as forming an $n$-letter "word" made up of $m$ $1$'s and the rest $0$'s.
Whichever way one views it, this can be done in $\dbinom{n}{m}$ ways.