Question

Determine the coefficient of $x_{1}^{3} x_{2} x_{3}^{4} x_{5}^{2}$ in the expansion of $$ \left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}\right)^{10} $$

Name: Problem 39, Chapter 5: Introductory Combinatorics
Uploaded: 2021-08-11T17:51:23-08:00
Duration: 3 min 58 s
Channel: Gaurav Kalra
Description: Determine the coefficient of x1^3 x2 x3^4 x5^2 in the expansion of (x1+x2+x3+x4+x5)^10

   Determine the coefficient of $x_{1}^{3} x_{2} x_{3}^{4} x_{5}^{2}$ in the expansion of
$$
\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}\right)^{10}
$$

Introductory Combinatorics

Richard A. Brualdi 5th Edition

Chapter 5, Problem 39

Instant Answer

Solved by Expert Gaurav Kalra

08/11/2021

Step 1

We can use the multinomial coefficient formula for this: $$\binom{10}{3, 1, 4, 0, 2} = \frac{10!}{3!1!4!0!2!} = \frac{10!}{3!4!2!} = \frac{10\times9\times8\times7\times6}{(3\times2)\times(4\times3\times2)\times(2)} = 10\times3\times7\times3 = 630$$ So there are Show more…

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Determine the coefficient of $x_{1}^{3} x_{2} x_{3}^{4} x_{5}^{2}$ in the expansion of $$ \left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}\right)^{10} $$

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Key Concepts

Multinomial Theorem

The multinomial theorem generalizes the binomial theorem for expanding expressions that are sums of more than two terms raised to a power. It states that when expanding (x1 + x2 + ... + xk)^n, the coefficient of any term x1^(a1) x2^(a2) ... xk^(ak) is given by the multinomial coefficient, provided that the exponents sum to n.

Multinomial Coefficient

A multinomial coefficient is a generalization of the binomial coefficient and is used to determine the number of ways to partition n objects into k groups with given sizes. It is computed using the formula n!/(a1! a2! ... ak!), where the sum of all a_i must equal n, and it directly gives the coefficient of the term in the expansion.

Factorial

Factorial, indicated by the symbol '!', is a product of all positive integers up to a given number. Factorials play a crucial role in combinatorics and the computation of coefficients in expansions because they count the number of arrangements of a set of items.

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