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Chapter 16: Problem 1

Explain why the expansion of \((x+y)^{n}=\sum_{i=0}^{n} C_{i} x^{n-i} y^{i}\) can also be written as \((x+y)^{n}=\sum_{i=0}^{n} C_{n-i} x^{n-i} y^{i}\)

Short Answer

Expert verified
The expressions are equivalent due to the property \( C_{i} = C_{n-i} \) (\( \binom{n}{i} = \binom{n}{n-i} \)).

Step by step solution

01

Understanding the Binomial Theorem

The Binomial Theorem states that for any positive integer \( n \), the expression \((x+y)^{n}\) can be expanded as the sum of terms of the form \( C_{i} x^{n-i} y^{i} \), where \( C_{i} \) is the binomial coefficient \( \binom{n}{i} \). The general term in this expansion is \( \binom{n}{i} x^{n-i} y^{i} \).
02

Reviewing the Terms of the Expansion

The expansion of \((x+y)^n\) is written as \( \sum_{i=0}^{n} C_{i} x^{n-i} y^{i} \). Here, each term has the form \( C_{i} x^{n-i} y^{i} \). Notice that the binomial coefficient \( C_{i} \) is actually \( \binom{n}{i} \).
03

Rearranging the Binomial Coefficient

We also know from the properties of binomial coefficients that \( \binom{n}{i} = \binom{n}{n-i} \). This equality holds because choosing \( i \) items from \( n \) is the same as leaving \( n-i \) items unchosen from \( n \).
04

Rewriting the Terms

By substituting \( C_{i} = \binom{n}{i} \) with \( \binom{n}{n-i} \), and recognizing they represent the same quantity, the expansion can be rewritten as \( \sum_{i=0}^{n} C_{n-i} x^{n-i} y^{i} \). Both expressions for the expansion are equivalent due to the symmetry in binomial coefficients.

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Key Concepts

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

Binomial Coefficient
The binomial coefficient is a fundamental component of the Binomial Theorem. It is denoted as \( \binom{n}{i} \) and is defined as the number of ways to choose \( i \) elements from a set of \( n \) elements without regard to the order of selection. This selection process forms the basis of combinatorics. The formula for calculating a binomial coefficient is \( \binom{n}{i} = \frac{n!}{i!(n-i)!} \), where \( n! \) denotes the factorial of \( n \).
  • "!" (factorial) means to multiply a series of descending natural numbers: for example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).
  • The role of the binomial coefficient is crucial in expanding an expression using the Binomial Theorem.
This symmetry of the binomial coefficient explains why \( \binom{n}{i} = \binom{n}{n-i} \), as choosing \( i \) elements to include also determines the \( n-i \) elements to exclude.
Algebraic Expansion
Algebraic expansion is a method used to simplify expressions that are raised to a power. The Binomial Theorem is a common technique employed to achieve this when dealing with binomials. For example, the binomial \((x+y)^n\) is expanded as a series of terms:\[ (x+y)^n = \sum_{i=0}^{n} \binom{n}{i} x^{n-i} y^{i} \]Each term in the expansion consists of a product of a binomial coefficient, a power of \( x \), and a power of \( y \). The exponents of \( x \) and \( y \) always add up to \( n \).
  • This expansion allows us to express complex powers of binomials as manageable polynomial expressions.
  • Each successive term decreases the power of \( x \) by one and increases the power of \( y \) by one.
This structure helps in simplifying computations or solving polynomial equations.
Combinatorics
Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination of objects. The Binomial Theorem's use of binomial coefficients is a prime example of combinatorial principles at work. In particular, the relationship \( \binom{n}{i} = \binom{n}{n-i} \) is a perfect illustration of how combinatorics can manifest symmetry.
  • Combinatorics helps in understanding permutations and combinations which are foundational to probability theory, number theory, and many aspects of algebra.
  • Using combinatorial thinking, we can solve problems related to finite structures, such as arranging students in a queue or distributing distinct items into boxes.
The strength of combinatorics within the Binomial Theorem is in its ability to provide precise mechanisms for calculating combinations, vital for scientific studies and complex calculations.
Mathematical Symmetry
Mathematical symmetry is a concept that suggests balance and beauty in mathematical structures. Within the context of the Binomial Theorem, symmetry refers to how the binomial coefficients \( \binom{n}{i} \) and \( \binom{n}{n-i} \) represent the same numbers because of their equal value.
  • Symmetry in mathematics is not merely about aesthetics but offers insights that simplify reasoning about equations and structures.
  • The symmetrical property of binomial coefficients is pivotal in transformations and simplifications of algebraic expressions.
This principle allows for alternative expressions of the Binomial Expansion formula, showing the universally flexible nature of mathematics. It aids mathematicians in visualizing equations and proofs in a new light, often leading to elegant solutions or reformulations.

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Most popular questions from this chapter

There are 3 seniors and 15 juniors in Mrs. Gillis's math class. Three students are chosen at random from the class. a. What is the probability that the group consists of a senior and two juniors? b. If the group consists of a senior and two juniors, what is the probability that Stephanie, a senior, and Jan, a junior, are chosen?

In \(3-6,\) find exact probabilities showing all required computation. A multiple-choice test of 10 questions has 4 choices for each question. Only one choice is correct. A student who did not study for the test guesses at each answer. a. What is the probability that that student will have exactly 5 correct answers? b. What is the probability that that student will have only 1 correct answer?

In \(3-6,\) find exact probabilities showing all required computation. A fair die is tossed five times. What is the probability of tossing a \(6 :\) $$ \begin{array}{llll}{\text { a. exactly once? }} & {\text { b. exactly twice? }} & {\text { c. exactly three times? }} \\ {\text { d. exactly four times? }} & {\text { e. exactly five times? }} & {\text { f. zero times? }}\end{array} $$ g. Which is the most likely event(s) when tossing a die five times?

In \(6-9,\) write an expression using sigma notation that can be used to find each probability. At most 7 tails when 10 coins are tossed

A machine purchased for \(\$ 75,000\) is expected to decrease in value by 20\(\%\) each year. The value of the machine after \(n\) years is \(75,000(1-0.20)^{n} .\) Use the binomial theorem to express the value of the machine after 5 years in sigma notation.
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