The numbers of terms in the expansion of ${\left(a+b+c\right)}^n$ , where $n\in N$ , is

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

Further expanding each term of R.H.S.

First term on expansion given one term

Second term on expansion gives two terms

Third term on expansion gives three terms and so on.

$\therefore$  Total no. of terms $=1+2+3+\mathrm{....}+\left(n+1\right)=\frac{\left(n+1\right)\left(n+2\right)}{2}$

Hence (A) is the correct answer.