Have you looked at this property?
So for example:
This may help a little.
FIRST, it must be pointed out that the is a mistake in the question.
This is Vandermonde's Theorem it should be
We can do a combinational proof.
is the number of ways to choose k items from m+n.
If then is the number of ways to choose those k items with j coming from the m group and k-j coming from the n group.
Now it us easy to see that
.