The number of terms in the expansion of a binomial

Question

If I have a binomial raised to the power n i.e $(a+b)^n$. Then the number of terms that its expansion yields will be $n+1$. This is very difficult for me to show(i.e How the number of terms are $n+1$). Any help will be appreciated. Tag is suggesgted.

Answer 1

You want to find the number of distinct terms of the form $a^kb^{n-k}$. Once you have picked a $k$, the exponent $n-k$ is set. For different $k_1$ and $k_2$, the terms $a^{k_1}$ and $a^{k_2}$ are different, so each $k$ defines a unique monomial. Since you can choose $k$ between $0$ and $n$, there are $n-0+1$ terms.

This is a direct approach. You can also prove that by induction.