Question

It \(\mathrm{A}\) and \(\mathrm{B}\) are coefficients of \(\mathrm{x}^{\mathrm{r}}\) and \(\mathrm{x}^{\mathrm{n}-\mathrm{r}}\) respectively in expansion of \((1+x)^{n}\) then \(=\) (a) \(\mathrm{A}+\mathrm{B}=\mathrm{n}\) (b) \(\mathrm{A}=\mathrm{B}\) (c) \(\mathrm{A}+\mathrm{B}=2^{\mathrm{n}}\) (d) \(A-B=2^{n}\)

Short answer

The correct option is (b) A = B, as the coefficients A and B of \(x^r\) and \(x^{n-r}\) terms in the expansion of \((1+x)^n\) are equal, i.e. \(A = \binom{n}{r} = \binom{n}{n-r} = B\).

Step-by-step solution

Recall the binomial expansion formula

Recall the binomial expansion formula for \((a+b)^n\), which can be written as: \[ (a+b)^n = \sum_{k=0}^n \binom{n}{k}a^{n-k}b^{k} \] Where \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) represents the binomial coefficients.

Apply the binomial expansion formula to our problem

Apply the binomial expansion formula to \((1+x)^n\): \[ (1+x)^n = \sum_{k=0}^n \binom{n}{k}1^{n-k}x^{k} = \sum_{k=0}^n \binom{n}{k}x^{k} \]

Find the coefficients A and B

We are given that A and B are the coefficients of \(x^r\) and \(x^{n-r}\) terms respectively. Let's locate these terms in the expansion: - The coefficient A is the coefficient of \(x^r\). In the expansion, this corresponds to the term \(\binom{n}{r}x^r\), so we have \(A = \binom{n}{r}\). - The coefficient B is the coefficient of \(x^{n-r}\). In the expansion, this corresponds to the term \(\binom{n}{n-r}x^{n-r}\), so we have \(B = \binom{n}{n-r}\).

Check the relationship between A and B

We have found that \(A = \binom{n}{r}\) and \(B = \binom{n}{n-r}\). Notice that by the property of binomial coefficients, \(\binom{n}{r} = \binom{n}{n-r}\), so we can conclude that \(A = B\).

Find the correct option

Given our analysis, we can see that the correct option is (b) A = B.

Key concepts

Binomial Theorem

The binomial theorem is a foundational principle in algebra that allows us to expand expressions raised to a power when they consist of two terms—a binomial—like \(a+b\). According to the theorem, the expansion of \(a+b)^n\), where \(n\) is a non-negative integer, can be expressed as a sum of terms in the form \(\binom{n}{k}a^{n-k}b^{k}\), where \(k\) ranges from 0 to \(n\). Each term includes a binomial coefficient \(\binom{n}{k}\), which is a specific number determining how many ways you can choose \(k\) elements from a set of \(n\) elements.

The binomial theorem is not just a mathematical novelty; it has practical applications in fields such as probability, statistics, and combinatorics. Understanding how to expand binomials provides insight into patterns of numbers and can be used to solve various problems in these disciplines.

Binomial Coefficients

Binomial coefficients, denoted as \(\binom{n}{k}\), are the numerical coefficients that appear in the binomial expansion. They play a central role by determining the weight of each term in the expansion. These coefficients also serve as a bridge between algebra and combinatorics since they represent the number of ways to choose \(k\) elements from a set of \(n\) elements, regardless of the order.

Computing a binomial coefficient involves factorial notation: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n!\) represents the factorial of \(n\), and it's the product of all positive integers up to \(n\). This relationship helps segue into the concept of factorials and their importance in combinatorics. When solving problems involving binomial expansion, recognizing patterns in these coefficients can greatly simplify the work.

Factorial Notation

Factorial notation is a convenient way to represent the product of an integer and all the positive integers below it. For any non-negative integer \(n\), the factorial is denoted by \(n!\) and is defined as \(n! = n \times (n-1) \times (n-2) \times ... \times 3 \times 2 \times 1\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). An important convention is that the factorial of 0, denoted as \(0!\), is 1.

Factorial notation is essential in combinatorics for calculating permutations and combinations—ways to arrange and select items, respectively. Understanding how factorials operate and their properties, such as \(n! = n \times (n-1)!\), is crucial in simplifying expressions involving binomial coefficients and solving combinatorial problems.

Combinatorics

Combinatorics is a branch of mathematics dealing with the counting, arrangement, and combination of objects. It provides tools for calculating the probabilities of various events and for finding the number of ways specific arrangements can be created. This discipline utilizes concepts like factorial notation and binomial coefficients in problems related to counting.

One common question in combinatorics is how many different ways can you select or arrange a subset of items from a larger set, a problem which can be solved using combinations and permutations. While permutations focus on the order of the elements, combinations do not. The problem of binomial expansion is inherently combinatorial, as each coefficient \(\binom{n}{k}\) in the expanded form represents a combination, telling us how many ways we can select \(k\) items from a set of \(n\), reflecting the profound interconnectedness of algebra and combinatorics.