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# Multinomial Theorem - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• 34 Questions around this concept.

## Solve by difficulty

The coefficient of $x^6$   in the expansion of $(1+x^2-x^3)^8$  is

If number of terms in the expansion of $(x-2 y+3 z)^n$ are 45, then n=

$(1+x)\left(1+x+x^2\right)\left(1+x+x^2+x^3\right) \ldots \ldots\left(1+x+x^2+\ldots \ldots+x^{100}\right)$ when written in the ascending power of x then the highest exponent of x is......

The number of terms in the expansion of $\small (a+b+c)^n$  will be:

## Concepts Covered - 1

Multinomial Theorem

As binomial has two terms raised to some power, similarly a multinomial has multiple terms raised to some power.

eg, $\mathrm{\left (x_1+x_2+x_3+........+x_k \right )}^n$      (Where n is a positive integer)

Expansion for the multinomial  $\mathrm{\left (x_1+x_2+x_3+........+x_k \right )}^n$ is

$\mathrm{\left ( x_1+x_2+x_3+.....+x_k \right )^n=\sum \frac{n!}{\alpha_1!\;\alpha_2!\; \alpha_3!\;...\;\alpha_k!}\mathrm{\left (x_1^{\alpha_1}\cdot x_2^{\alpha_2}\cdot x_3^{\alpha_1}\cdot ....\cdot x_k^{\alpha_k} \right )}}$

with α1 + α2 + α3 + …. + α= n,

and α1, α2, α3, …., αk  $\in W$

In particular,

$\mathrm{\left(a+b+c\right)^n=\sum \frac{n!}{\left(\alpha !\right)\left(\beta !\right)\left(\gamma !\right)}a^{\alpha }\:b^{\beta }\:c^{\gamma }}$

The number of distinct terms in the multinomial expansion $\mathrm{\left (x_1+x_2+x_3+........+x_k \right )}^n$ is  n + k - 1Ck - 1

## Study it with Videos

Multinomial Theorem

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