<s>[INST] Number of terms in a Polynomial Expansion 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:


*

*What is the number of terms for $(a + b + c + d + e)^n$?

*Do we have a general formula for the number of terms of a polynomial expansion?

*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?

*What if the given is $(a + b + Constant)^n$, would the constant affect the number of terms?

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