Problem 4 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If the coefficients of rth term and \((r+4)\) th term are equal in the expansion of \((1+x)^{20}\), then the value of \(r\) will be (a) 7 (b) 8 (c) 9 (d) 10

Step-by-Step Solution

Answer

The value of \(r\) is 9.

Step 1: Understand the Problem

We are given an expansion of \((1+x)^{20}\) and need to find the value of \(r\) where the coefficients of the \(r\)th term and the \((r+4)\)th term are equal.

Step 2: Use Binomial Theorem

According to the binomial theorem, the general term \(T_k\) in the expansion of \((1+x)^{20}\) is given by \(\binom{20}{k-1}x^{k-1}\). Here, the coefficient of the \(r\)th term is \(\binom{20}{r-1}\) and the coefficient of the \((r+4)\)th term is \(\binom{20}{r+3}\).

Step 3: Set Up the Equation

According to the given condition, the coefficients of the \(r\)th term and the \((r+4)\)th term are equal. Therefore, \(\binom{20}{r-1} = \binom{20}{r+3}\).

Step 4: Use Property of Binomial Coefficients

We know \(\binom{n}{k} = \binom{n}{n-k}\). Here, it means \(\binom{20}{r+3} = \binom{20}{20-(r+3)} = \binom{20}{17-r}\). Now we have \(\binom{20}{r-1} = \binom{20}{17-r}\).

Step 5: Solve the Equation

The equation \(\binom{20}{r-1} = \binom{20}{17-r}\) implies that \(r-1 = 17-r\). By solving this equation, \(2r = 18\), so \(r = 9\).

Step 6: Finalize the Answer

Thus, the value of \(r\) that satisfies the condition is 9.

Key Concepts

Binomial Expansion

The binomial expansion is a powerful mathematical concept used to expand expressions that are raised to a power. It is particularly applicable to expressions of the form \((a + b)^n\). The Binomial Theorem provides a formula to expand such expressions in terms of their individual powers of \(a\) and \(b\). In the context of the problem at hand, we're dealing with the binomial expression \((1+x)^{20}\). This means we need to find the terms in the expansion of this expression. Each term in the expansion of a binomial can be expressed using binomial coefficients, which are derived from Pascal's Triangle or calculated using a factorial formula. The general expression for any term in a binomial expansion is given by: \[ T_k = \binom{n}{k} a^{n-k} b^k \] For \((1+x)^{20}\), this simplifies to: \[ T_k = \binom{20}{k} x^k \] where \(k\) varies from 0 to 20. Identifying specific terms and their coefficients is central to problems involving binomial expansion.

Binomial Coefficients

Binomial coefficients are numerical factors that multiply the terms in a binomial expansion. They are denoted as \(\binom{n}{k}\), representing "\(n\) choose \(k\)" and calculated using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \(n!\) (n factorial) is the product of all positive integers up to \(n\). In the context of \((1+x)^{20}\), the coefficients determine the weight each power of \(x\) has in the expansion. In our specific problem, we looked at the coefficients of the \(r\)th term and the \((r+4)\)th term, namely \(\binom{20}{r-1}\) and \(\binom{20}{r+3}\). Due to the problem's condition, these two coefficients are equal. We utilized a critical property of binomial coefficients that \(\binom{n}{k} = \binom{n}{n-k}\) to simplify and solve for \(r\). Understanding and manipulating these coefficients derives the solution in binomial expansion problems.

Combinatorics

Combinatorics is a branch of mathematics focused on counting and arrangement possibilities, often dealing with combinations and permutations. Binomial coefficients are deeply linked to combinatorics, as they represent the number of ways to choose \(k\) items from \(n\) without regard to order. When solving binomial expansion problems, understanding how to apply combinatorial principles is key. This involves recognizing how the coefficients correspond to combinations, which in our problem is revealed when we equate the coefficients \(\binom{20}{r-1}\) and \(\binom{20}{r+3}\). The property \(\binom{n}{k} = \binom{n}{n-k}\), a combinatorial symmetry, helps us find the correct terms to compare. In solving our problem, we recognized that combinatorics provides the tools to handle symmetry and balance within the binomial expansion. These principles helped us deduce that \(r\) must equal 9 to satisfy the equal coefficients condition, showcasing the underlying elegance of combinatorial math.