Question

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

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

Answer 1

The correct option is D

$66$

Calculate the number of terms in the expansion of the given expression

The given expression is ${\left(a+b+c\right)}^{10}$.

According to the binomial expansion,

The number of terms in the expansion of ${\left(a_1+a_2+...+a_k\right)}^n$ can be calculated by the following formula,

Total number of terms $={}^{n+k-1}C_{k-1}$

Now, in the given expression ${\left(a+b+c\right)}^{10}$,

$$\begin{gathered} n=10 \\ k=3 \end{gathered}$$

The number of terms in the given expression can be calculated as,

$\because$ Number of terms $={}^{n+k-1}C_{k-1}$

$\Rightarrow$ Number of terms $={}^{10+3-1}C_{3-1}$

$\Rightarrow$ Number of terms $={}^{12}C_2$

$\Rightarrow$ Number of terms $=\frac{12!}{2!\left(12-2\right)!}$ $\left[\because {}^{n}C_r=\frac{n!}{r!\left(n-r\right)!}\right]$

$\Rightarrow$ Number of terms $=\frac{12\times 11\times 10!}{2\times 1\times 10!}$

$\Rightarrow$ Number of terms $=66$

Hence, option (D) is the correct option.