Number of terms in a Polynomial Expansion

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For a binomial $(a + b)^n$, the number of terms is n + 1.

For a trinomial $(a + b + c)^n$, the number of terms is $\frac{(n+1)(n+2)}{(2)}$.

For a multinomial $(a + b + c +d)^n$, the number of terms is $\frac{(n+1)(n+2)(n+3)}{(6)}$.

I'm guessing that for $(a + b + c + d + e)^n$, the number of terms formula would include $(n+1)(n+2)(n+3)(n+4)$ on the numerator but I don't know what should be its denominator.

Question:

1. What is the number of terms for $(a + b + c + d + e)^n$?
2. Do we have a general formula for the number of terms of a polynomial expansion?
3. What if the given is $(a^2 + a + b)^n$, can I still use the formula $\frac{(n+1)(n+2)}{(2)}$.when 2 terms in the expansion has the same variable?
4. What if the given is $(a + b + Constant)^n$, would the constant affect the number of terms?

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2 Answers

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For a multinomial $(\sum_{i=1}^m x_i)^n$, the number of terms are ${{n+m-1} \choose {m-1}}$ or ${{n+m-1} \choose {n}}$

This is the same as n apples in m baskets. The combinations are ${{n+m-1} \choose {m-1}}$ or ${{n+m-1} \choose {n}}$

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for (a+b+c+d+e)^n it would be [(n+4)(n+3)(n+2)(n+1)]/24

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