Find the number of terms in the expansion of

We have:

$(a+b+c)^{n}=[a+(b+c)]^{n}$

$=a^{n}+{ }^{n} C_{1} a^{n-1}(b+c)^{1}+{ }^{n} C_{2} a^{n-2}(b+c)^{2}+\ldots+{ }^{n} C_{n}(b+c)^{n}$

Further, expanding each term of R.H.S., we note that

First term consists of 1 term.

Second term on simplification gives 2 terms.

Third term on expansion gives 3 terms.

Similarly, fourth term on expansion gives 4 terms and so on.

$\therefore$ The total number of terms $=1+2+3+\ldots .+(n+1)=\frac{(n+1)(n+2)}{2}$