Question

a. If we applied the pattern of the coefficients to the coefficient of the first term in a binomial expansion, the coefficient would be $\frac{n !}{0 !(n-0) !}$. Show that this expression is 1. b. If we applied the pattern of the coefficients to the coefficient of the last term in a binomial expansion, the coefficient would be $\frac{n !}{n !(n-n) !} .$ Show that this expression is 1.

Name: Problem 103, Chapter 14: Elementary and Intermediate Algebra
Uploaded: 2020-04-11T16:14:27-08:00
Duration: 59 s
Channel: Heather Zimmers
Description: a. If we applied the pattern of the coefficients to the coefficient of the first term in a binomial expansion, the coefficient would be (n !)/(0 !(n-0) !). Show that this expression is 1. b. If we applied the pattern of the coefficients to the coefficient of the last term in a binomial expansion, the coefficient would be (n !)/(n !(n-n) !) . Show that this expression is 1.

   a. If we applied the pattern of the coefficients to the coefficient of the first term in a binomial expansion, the coefficient would be $\frac{n !}{0 !(n-0) !}$. Show that this expression is 1.
b. If we applied the pattern of the coefficients to the coefficient of the last term in a binomial expansion, the coefficient would be $\frac{n !}{n !(n-n) !} .$ Show that this expression is 1.

Elementary and Intermediate Algebra

Alan S. Tussy, R.… 5th Edition

Chapter 14, Problem 103

Instant Answer

Solved by Expert Heather Zimmers

04/11/2020

Step 1

We know that 0! is defined to be 1. So, we can simplify the expression to $\frac{n !}{1*(n-0) !}$. Show more…

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a. If we applied the pattern of the coefficients to the coefficient of the first term in a binomial expansion, the coefficient would be $\frac{n !}{0 !(n-0) !}$. Show that this expression is 1. b. If we applied the pattern of the coefficients to the coefficient of the last term in a binomial expansion, the coefficient would be $\frac{n !}{n !(n-n) !} .$ Show that this expression is 1.

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Key Concepts

Factorial

A factorial, denoted by n!, is the product of all positive integers up to n. Factorials are essential in the definition of binomial coefficients. They also play a crucial role in various areas of mathematics, including combinatorics, probability, and algebra.

Binomial Coefficients

Binomial coefficients represent the coefficients in the expansion of a binomial expression raised to a power, and they are given by the formula n! / (k!(n-k)!) for the kth term. This formula is fundamental in combinatorics, as it not only provides the coefficients in the expansion but also counts the number of ways to choose k elements from a set of n elements.

Definition of 0!

The convention that 0! equals 1 is critical for ensuring that binomial coefficient formulas, such as when k = 0 or k = n, yield consistent and correct results. This definition supports the idea that there is exactly one way to choose nothing (or everything) from a set.

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