Multinomial Theorem - Practice Questions & MCQ

As a binomial has two terms raised to some power, similarly a multinomial has multiple terms raised to some power.
$\mathrm{eg}_{,}\left(\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\ldots \ldots .+\mathrm{x}_{\mathrm{k}}\right)^n \quad($ Where n is a positive integer)
Expansion for the multinomial $\left(\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\ldots \ldots . .+\mathrm{x}_{\mathrm{k}}\right)^n$ is

$
\left(\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\ldots .+\mathrm{x}_{\mathrm{k}}\right)^{\mathrm{n}}=\sum \frac{\mathrm{n}!}{\alpha_{1}!\alpha_{2}!\alpha_{3}!\ldots \alpha_{\mathrm{k}}!}\left(\mathrm{x}_1^{\alpha_1} \cdot \mathrm{x}_2^{\alpha_2} \cdot \mathrm{x}_3^{\alpha_1} \cdot \ldots \cdot \mathrm{x}_{\mathrm{k}}^{\alpha_{\mathrm{k}}}\right)
$

with $a_1+a_2+a_3+\ldots .+a_k=n$,
and $a_1, a_2, a_3, \ldots, a_k \in W \mid$
In particular,

$
(\mathrm{a}+\mathrm{b}+\mathrm{c})^{\mathrm{n}}=\sum \frac{\mathrm{n!}}{(\alpha!)(\beta!)(\gamma!)} \mathrm{a}^\alpha \mathrm{b}^\beta \mathrm{c}^\gamma
$
The number of distinct terms in the multinomial expansion $\left(\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\ldots \ldots .+\mathrm{x}_{\mathrm{k}}\right)^{\mathrm{n}}{ }_{\text {is }}{ }^{\mathrm{n}+\mathrm{k}-1} \mathrm{C}_{\mathrm{k}-1}$