Question

Find the number of terms in the expansion of $(a+b+c{)}^n.$

Answer 1

A is the correct choice. We have $(a+b+c{)}^n=[a+(b+c)]n$

$=a^n{+}^n{C}_1{a}^{n-1}(b+c{)}^1{+}^n{C}_2{a}^{n-2}(b+c{)}^2+.....{+}^n{C}_n(b+c{)}^n$

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

First term consist 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.

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