Problem 82 - Algebra and Trigonometry

Question

Why Is \(\left(x^{n}\right)\) the Same as \(C(n, r) ?\) This exercise explains why the binomial coefficients \(\left(\begin{array}{c}{n} \\\ {r}\end{array}\right)\) that appear in the expansion of \((x+y)^{n}\) are the same as \(C(n, r),\) the number of ways of choosing \(r\) objects from \(n\) objects. First, note that expanding a binomial using only the Distributive Property gives $$ \begin{aligned}(x+y)^{2} &=(x+y)(x+y) \\ &=(x+y) x+(x+y) y \\ &=x x+x y+y x+y y \\\\(x+y)^{3} &=(x+y)(x x+x y+y x+y y) \\ &=x x x+x x y+x y x+x y y+y x x \\ &+y x y+y y x+y y y \end{aligned} $$ (a) Expand \((x+y)^{5}\) using only the Distributive Property. (b) Write all the terms that represent \(x^{2} y^{3}\) together. These are all the terms that contain two \(x^{\prime}\) s and three \(y^{\prime} s .\) (c) Note that the two \(x\) 's appear in all possible positions. Conclude that the number of terms that represent \(x^{2} y^{3}\) is \(C(5,2) .\) (d) In general, explain why \((r)\) in the Binomial Theorem is the same as \(C(n, r) .\)

Step-by-Step Solution

Answer

We use combinations to select positions for \(x\) in the polynomial, represented by \(C(n, r)\), which is the same as the binomial coefficients.

Step 1: Expand (x+y)^5

To expand \((x+y)^5\), we apply the Distributive Property repeatedly:\[(x+y)^5 = (x+y)(x+y)(x+y)(x+y)(x+y)\]First, expand \((x+y)(x+y)\) to get:\[(x+y)^2 = x^2 + xy + yx + y^2\]Similarly, expand the resultant terms further by multiplying them with \((x+y)\) repeatedly until you reach the fifth power:\[(x+y)^3 = (x^2 + xy + yx + y^2)(x+y)\]Next:\[(x+y)^4 = (x^3 + 3x^2y + 3xy^2 + y^3)(x+y)\]Finally:\[(x+y)^5 = (x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4)(x+y)\]

Step 2: Group Terms with x^2 y^3

From the expansion of \((x+y)^5\), identify the terms that contribute to \(x^2 y^3\): - All terms of the expansion should be analyzed to see how each contributes \(x\) and \(y\).- There are terms such as \(xxyyy, xyxyy, xyxxy, xyyxy, xyyxy,\) and so forth. We rearrange the exponents that total to the expression \(x^2 y^3\).

Step 3: Determine Combinations

Realize that to form \(x^2 y^3\), we must choose positions for \(x\)'s among the 5 positions (since the expanded result has 5 bases: \(x or y\)).- This is a combination problem of choosing 2 positions for \(x\) out of 5, which is computed using binomial coefficient formula:\[C(5,2) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10\]So, there are 10 distinct terms that express \(x^2 y^3\).

Step 4: Generalization to C(n, r)

The reasoning used to identify \(x^2 y^3\) can be generalized for any expansion of \((x+y)^n\).- When dealing with \((x+y)^n\), to get \(x^r y^{n-r}\), select \(r\) positions for \(x\) out of \(n\) total positions.- This selection is given by the binomial coefficient \(C(n, r)\), which matches the term in the Binomial Theorem expansion \( (x + y)^n = \sum_{r=0}^{n} C(n, r) x^r y^{n-r} \).- Thus, both numerically and conceptually, \(C(n, r)\) relates to selecting terms in the expansion and combinations of positions.

Key Concepts

Distributive Property

The Distributive Property is a foundational concept in algebra that allows us to simplify expressions by breaking them down into smaller parts, making multiplication easier. Essentially, it tells us how to handle scenarios where we have to multiply a sum by another number or variable.

For the binomial expression \(x + y\), the Distributive Property becomes especially useful when we're expanding expressions such as \( (x+y)^n \). Let's illustrate this with a simple product: \( (x+y)^2 \). By distributing carefully, we have \( (x+y)(x+y) = x \cdot x + x \cdot y + y \cdot x + y \cdot y \).

This expansion lets us see each term separately, allowing easier computation of longer expressions like \( (x+y)^5 \). Each time you multiply by another \( (x+y) \), you're again applying the Distributive Property iteratively, which uncovers all possible combinations of the multiplicands x and y.

Utilizing this property repeatedly is what leads to the orderly expansion of binomial terms, eventually leading us to discover patterns involving the binomial coefficients.

Binomial Coefficient

A binomial coefficient appears as an essential component in binomial expansions, represented mathematically as \( \binom{n}{r} \). It signifies the number of ways you can choose \( r \) items from \( n \) total items, a concept crucial in both algebra and combinatorics.

The significance of binomial coefficients can be clearly seen in the Binomial Theorem, which expresses the expansion of \( (x+y)^n \). In this expression, the term \( \binom{n}{r} x^r y^{n-r} \) appears and reflects both the concept of choosing \( r \) positions for \( x \) among \( n \) trials, as well as the number of different ways to arrange those elements.

From computations previously shown, we learned that \( \binom{5}{2} \), for instance, gives us 10—indicating 10 possible arrangements of \( x^2y^3 \) in the expansion of \( (x+y)^5 \). Each coefficient directly links to a combination, pointing out in how many distinct ways each term can appear in the expanded expression.

Combinatorics

Combinatorics is the field of mathematics that focuses on counting, arranging, and finding patterns. Understanding how to calculate combinations lies at the heart of this field, which involves selecting items from a larger set without regard to the order of selection.

When working through a problem like expanding \( (x+y)^5 \), you're engaging with prioritizing the selection and arrangement of terms systematically, a task under combinatorics. The problem of expanding binomials quickly turns into a question of how many different ways we can choose positions for each variable, rendering the need to comprehend the interplay between arrangements and placements.

In every combination task—like placing \( r \) \( x\)'s within \( n \) total slots—the binomial coefficient \( C(n, r) \) depicts the number of possibilities available. It's an application of combinatorics to a practical scenario within algebraic operations, providing a structured way to evaluate expansions through precise counts rather than trial and error.