The number of distinct terms in the expansion of  ${\left(x_1+x_2+\cdots +x_n\right)}^3$ is

 

We know that

$\begin{array}{r}{\left(x_1+x_2+\cdots +x_n\right)}^3\\ =\sum \frac{3!}{{\lambda }_1!{\lambda }_2!\cdots {\lambda }_n!}{x}_1^{\lambda 1}{x}_2^{\lambda 2}\cdots x_n^{\lambda n}\end{array}$

where ${\lambda }_1+{\lambda }_2+\cdots +{\lambda }_n=3$ and ${\lambda }_i\ge 0$

Thus, number of distinct terms in the expansion of

${\left(x_1+x_2+\cdots +x_n\right)}^3$

is equal to the number of non-nagative integral solution of 

${\lambda }_1+{\lambda }_2+\cdots +{\lambda }_n=3$

= number of ways of distributing 3 identical objects among n persons

${=}^{3+n-1}{C}_3{=}^{n+2}C3$