/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Give a combinatorial interpretat... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 21

Give a combinatorial interpretation of the coefficient of \(x^{4}\) in the expansion \(\left(1+x+x^{2}+x^{3}+\cdots\right)^{3} .\) Use this interpretation to find this number.

Short Answer

Expert verified
The coefficient of \(x^4\) is 15.

Step by step solution

01

- Understand the series expansion

The series \(1 + x + x^2 + x^3 + \cdots\) can be rewritten as a geometric series: \(\frac{1}{1-x}\). So, \(\left(1+x+x^2+x^3+\cdots\right)^3\) can be rewritten as \(\left( \frac{1}{1-x} \right)^3\).
02

- Rewrite using binomial series

Using the binomial series, \(\left( \frac{1}{1-x} \right)^3 = \sum_{n=0}^{\infty} \binom{n+2}{2} x^n \). Identify that each term in the expansion represents the combination of summing up powers of \( x \).
03

- Find the coefficient of \(x^4\)

We need the coefficient of \(x^4\). From \(\sum_{n=0}^{\infty} \binom{n+2}{2} x^n\), the coefficient of \(x^4\) is \(\binom{4+2}{2}\).
04

- Calculate the binomial coefficient

Calculate \(\binom{4+2}{2} = \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric series
A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In other words, a geometric series can be written as:
  • First term: \( a \)
  • Second term: \( ar \)
  • Third term: \( ar^2 \)
and so on. The series continues indefinitely. In our case, the series given is \(1 + x + x^2 + x^3 + \, \ldots \). This can be recognized as a geometric series where the first term \( a \) is 1, and the common ratio \( r \) is \( x \). Using the formula for the sum of an infinite geometric series, we convert it into \( \frac{1}{1-x} \). This helps in simplifying the expression with combinatorial techniques.
Binomial series
The binomial series is an expansion of the form \((1 + x)^n\) into powers of \(x\). For any real number \(n\), the series is given by:
  • \( \sum_{k=0}^\infty \binom{n}{k} x^{k} \)
In our problem, we are dealing with the expression \( \left( \frac{1}{1-x} \right)^3 \), which can be expanded using the binomial series. Specifically, it takes the form:
  • \( \sum_{n=0}^\infty \binom{n+2}{2} x^n \)
Here, we are interested in finding the coefficient of \( x^4 \) within this expansion. Recognizing and using the binomial series makes it easier to identify the coefficients of the terms in the expanded series.
Binomial coefficient
The binomial coefficient, typically written as \( \binom{n}{k} \), represents the number of ways to choose \( k \) items from \( n \) items without regard to the order of selection. It's also known as a combination. Mathematically, it is given by:
  • \( \binom{n}{k} = \frac{n!}{k! \cdot (n-k)!} \)
In our exercise, the task is to find the coefficient of \( x^4 \) in the expansion. From the binomial series step, the coefficient of \( x^4 \) corresponds to \( \binom{4 + 2}{2} \), or simply \( \binom{6}{2} \). Calculating this, we get:
  • \( \binom{6}{2} = \frac{6!}{2! \cdot (6 - 2)!} = \frac{6 \times 5}{2 \times 1} = 15 \)
Therefore, the coefficient of \( x^4 \) in the expansion is 15. Understanding how to calculate binomial coefficients is crucial for breaking down and solving combinatorial problems effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use generating functions to solve the recurrence relation \(a_{k}=5 a_{k-1}-6 a_{k-2}\) with initial conditions \(a_{0}=6\) and \(a_{1}=30\)

a) Find a recurrence relation for the number of ways to completely cover a \(2 \times n\) checkerboard with \(1 \times 2\) dominoes. [Hint: Consider separately the coverings where the position in the top right corner of the checkerboard is covered by a domino positioned horizontally and where it is covered by a domino positioned vertically.] b) What are the initial conditions for the recurrence relation in part (a)? c) How many ways are there to completely cover a \(2 \times\) 17 checkerboard with \(1 \times 2\) dominoes?

Show that the Fibonacci numbers satisfy the recurrence relation \(f_{n}=5 f_{n-4}+3 f_{n-5}\) for \(n=5,6,7, \ldots,\) together with the initial conditions \(f_{0}=0, f_{1}=1, f_{2}=1, f_{3}=2\) and \(f_{4}=3 .\) Use this recurrence relation to show that \(f_{5 n}\) is divisible by \(5,\) for \(n=1,2,3, \ldots\)

(Requires calculus) Show that if \(G_{X}\) is the probabili generating function for a random variable \(X\) such the \(X(s)\) is a nonnegative integer for all \(s \in S\) , then a) \(G_{X}(1)=1 .\) b) \(E(X)=G_{X}^{\prime}(1)\) c) \(V(X)=G_{X}^{\prime \prime}(1)+G_{X}^{\prime}(1)-G_{X}^{\prime}(1)^{2}\)

Dynamic programming can be used to develop an algorithm for solving the matrix-chain multiplication problem introduced in Section \(3.3 .\) This is the problem of determining how the product \(\mathbf{A}_{1} \mathbf{A}_{2} \cdots \mathbf{A}_{n}\) can be computed using the fewest integer multiplications, where \(\mathbf{A}_{1}, \mathbf{A}_{2}, \ldots, \mathbf{A}_{n}\) are \(m_{1} \times m_{2}, m_{2} \times m_{3}, \ldots, m_{n} \times m_{n+1}\) matrices, respectively, and each matrix has integer entries. Recall that by the associative law, the product does not depend on the order in which the matrices are multiplied. a) Show that the brute-force method of determining the minimum number of integer multiplications needed to solve a matrix-chain multiplication problem has exponential worst-case complexity. [Hint: Do this by first showing that the order of multiplication of matrices is specified by parenthesizing the product. Then, use Example 5 and the result of part (c) of Exercise 43 in Section 8.4 .1 b) Denote by \(A_{i j}\) the product \(\mathbf{A}_{i} \mathbf{A}_{i+1} \ldots, \mathbf{A}_{j}\) and \(M(i, j)\) the minimum number of integer multiplications required to find \(\mathbf{A}_{i j} .\) Show that if the least number of integer multiplications are used to compute \(\mathbf{A}_{i j},\) where \(i
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.