What is the coefficient of the term $a^i b^j c^k$ in the expansion of $(a+b+c)^n$ ?
Given solution:
Before collecting like terms, each term in the expansion is a product of $n$ factors each of which is $a, b,$ or $c$. Those terms which have exactly i a’s, j b’s and k c’s can be combined into the single term $a^i b^j c^k$ and the number of such terms will be equal to the number of n letter strings containing i a’s, j b’s and k c’s, which by the Mississippi rule is $\frac{n!}{(i!\times j! \times k!)}$.
I understand that this problem is actually about the derivation of the coefficient of the multinomial theorem,but I am not sure about the combinatoric reasoning,what I mean is that I understand the analogy problem but what I couldn't is how the analogy holds here,Could anybody explain this to me?