Show that the coefficient of x^n in the expansion of (1+x)^2 n is (2 n) ! /(n !)^2.

Name: Problem 51, Chapter 14: Precalculus
Uploaded: 2020-07-13T20:43:41-08:00
Duration: 2 min 52 s
Channel: Grant Mansfield
Description: Show that the coefficient of x^n in the expansion of (1+x)^2 n is (2 n) ! /(n !)^2.

step 1 All right, so for this question, we are asked to show that the coefficient of x to the power of

Show that the coefficient of x^n in the expansion of (1+x)^2...

Key Concepts

Binomial Theorem

The Binomial Theorem provides a way to expand expressions of the form (a + b)^n as a sum of terms involving binomial coefficients. It states that (a + b)^n equals the sum from k = 0 to n of C(n, k) a^(n-k) b^k, where C(n, k) represents the number of ways to choose k items from a set of n, which directly links the theorem to combinatorial counting methods.

Binomial Coefficient

A binomial coefficient, typically written as C(n, k) or (n choose k), represents the number of ways to choose k elements from a set of n elements without regard to the order of selection. It is calculated using the formula n!/(k!(n-k)!), where n! denotes the factorial of n. This coefficient directly appears in the expansion given by the Binomial Theorem.

Factorial Notation

Factorial notation, represented by an exclamation mark after a number (n!), denotes the product of all positive integers up to that number. Factorials are fundamental in counting and combinatorics, particularly in the computation of binomial coefficients since they help in determining the number of permutations or combinations of a given set.

Coefficient Extraction in Binomial Expansion

In the context of a binomial expansion, each term’s coefficient is determined by the binomial coefficient corresponding to the indices of the terms in the sum. Extracting the coefficient of a specific power of x involves identifying the term in the expansion whose power matches the desired exponent, and then evaluating the associated binomial coefficient expression.

Transcript

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