Suppose we have 'n' Stars and 'k' bins. We have to distribute 'n' stars in 'k' bins (using k-1 bars) such that bins can be empty. We can do this in $\binom {n+k-1}{k-1}$ = $\binom {n+k-1}{n}$ ways. But Sometimes we use another notation $\left({{k}\choose {n}}\right)$ to represent $\binom {n+k-1}{k-1}$, Which says 'k multichoose n'. But normally as n > k, then how can we choose more objects from less objects i.e. n from k. So, I think It has to be like this $\left({{n}\choose {k}}\right)$.
Combinatorics – How to Use Multichoosing (Stars and Bars)
combinatorics
Best Answer
The notation $\left( \! \binom{k}{n} \! \right)$ stands for the number of multisets of size $n$ on $k$ symbols. The important thing to note is that the $k$ symbols are distinguishable and the $n$ "positions" in the multiset are indistinguishable (order doesn't matter).
This suggests a bijection (one-to-one correspondence) with the number of ways to put balls (indistinguishable objects) in labeled boxes (distinguishable objects). Specifically, the $k$ symbols in the multiset correspond to the boxes, and the $n$ positions in the multiset correspond to the balls.
Therefore, $$ \left( \! \binom{k}{n} \! \right) = \binom{n + k - 1}{k - 1} = \binom{n + k - 1}{n}. $$ The only restrictions on the integers $k$ and $n$ are $$ n \ge 0 \quad \text{and} \quad k \ge 1. $$
In particular, it doesn't matter which is greater.