Question

The number of terms in the expansion of${(a+b+c)}^n$, where $n\in N$ .

Answer 1

Find the number of terms.

The total number of terms $\left(N\right)$ in the expansion of ${(a_1+a_2+a_3+...+a_k)}^n$ is given by, ${}^{n+k-1}C_{k-1}$.

We have to find the number of terms in the expansion of${(a+b+c)}^n$.

Here $k=3$

$$\begin{gathered} {}^{n+3-1}C_{3-1}={}^{n+2}C_2 \\ \Rightarrow =\frac{(n+2)!}{2!(n+2-2)!} \\ \Rightarrow =\frac{(n+2)!}{2(n!)} \\ \Rightarrow =\frac{(n+2)(n+1)n!}{2·n!} \\ \Rightarrow =\frac{(n+2)(n+1)}{2} \end{gathered}$$

Hence, the number of terms is $\frac{(n+1)(n+2)}{2}$.