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Multinomial Theorem - Practice Questions & MCQ

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

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  • 34 Questions around this concept.

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QuestionMEDIUM

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

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

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(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......

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

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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 + …. + αk = 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

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Multinomial Theorem

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Multinomial Theorem Current Topic

Binomial Theorem - Formula, Expansion, Problems and Applications
Some Standard Expansions
General and Middle Terms
Greatest Term (numerically)
An Important Theorem
Problems on Divisibility
Finding last digits
Important Result (Comparison)
Multinomial Theorem
Series Involving Binomial Coefficients

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