Question

The number of terms in the expansion of $(a+b+c{)}^{10}$ is

• A. $11$
• B. $21$
• C. $55$
• D. $66$

Answer 1

Answer: (D)

We have, $(a+b+c{)}^{10}=[a+(b+c){]}^{10}$
$={}^{10}{C}_0{a}^{10}+{}^{10}{C}_1{a}^9\left(b+c\right)+{}^{10}{C}_2{a}^8{\left(b+c\right)}^2+\dots$
$\dots +{}^{10}{C}_{10}{\left(b+c\right)}^{10}$
So, we can see that first term contains $1$ term, second term contains $2$ terms, $3^{rd}$ term contains $3$ terms and so on.
$\therefore$ Total number of terms in the given expansion
$=1+2+3+...+11$
$=\frac{11\left(11+1\right)}{2}=66$