Given $a,b,c,n\in\mathbb{N}^+$, and $a,b<c$, in which conditions does the relation
$$ \binom{a+n-1}{n}+\binom{b+n-1}{n}\geq\binom{c+n-1}{n} $$
hold?
Since I am not an expert in this field, I beg your pardon in case the question is too trivial (or too complex)! Thanks for your help!
EDIT: The problem is more appealing if we further suppose $a+b>c$.
Let $ n=1$, then $$\binom{a+n-1}{n}+\binom{b+n-1}{n}\geq\binom{c+n-1}{n}\iff a+b\ge c $$
which does not have to be true.
Thus $$a\le b<c $$ is not sufficient for $$\binom{a+n-1}{n}+\binom{b+n-1}{n}\geq\binom{c+n-1}{n}$$