What is the coefficient of x1^3 x2^3 x3 x4^2 in the expansion of (x1-x2+2 x3-2 x4)^9 ?

Name: Problem 40, Chapter 5: Introductory Combinatorics
Uploaded: 2020-07-17T18:19:06-08:00
Duration: 2 min 11 s
Channel: Linh Vu
Description: What is the coefficient of x1^3 x2^3 x3 x4^2 in the expansion of (x1-x2+2 x3-2 x4)^9 ?

step 1 In this question, we're given the 2x plus 1 over x square, and then 3 power 6 by the binomial th

What is the coefficient of x1^3 x2^3 x3 x4^2 in the...

Key Concepts

Multinomial Coefficients

Multinomial coefficients arise in the expansion provided by the multinomial theorem. They are the generalization of binomial coefficients and are computed as n! divided by the product of the factorials of the exponents of each term. These coefficients determine the number of ways to choose specified numbers of occurrences for each term in the expanded expression.

Sign and Constant Factor Consideration

When terms in the polynomial have associated constants or negative signs, it is essential to incorporate these into the calculation of the coefficient. Each instance of a term with a negative sign or constant factor will contribute its power repetition to the final coefficient. Thus, the overall sign and magnitude are determined by raising the constants to the required exponents and then multiplying them by the multinomial coefficient.

Multinomial Theorem

The multinomial theorem generalizes the binomial theorem for expanding powers of a sum of more than two terms. It states that the expansion of (a1 + a2 + ... + ak)^n is a sum over all possible combinations of non-negative integers k1, k2, ..., kk that sum to n, with each term having the multinomial coefficient n!/(k1! k2! ... kk!) multiplied by the corresponding products a1^k1 * a2^k2 * ... * ak^kk.

Coefficient Calculation in Polynomial Expansions

Calculating the coefficient of a specific term in a polynomial expansion involves identifying the term's exponents for each variable and then using the appropriate multinomial coefficient along with any additional factors arising from the term's coefficients and signs. This process involves matching the given exponents with those in the expansion and accounting for the constants multiplying each variable.

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