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

Question

If the coefficient of 7 th and 13 th term in the expansion of \((1+x)^{n}\) are equal, then \(n=\) (a) 10 (b) 15 (c) 18 (d) 20

Step-by-Step Solution

Answer

The value of \(n\) is 18.

Step 1: Formula for Binomial Coefficient

In the binomial expansion of \((1+x)^n\), the coefficient of the \(r^{th}\) term is given by the binomial coefficient \( \binom{n}{r-1} \). For this problem, we need to compare two different coefficients.

Step 2: Expression for Coefficients

The coefficient of the \(7^{th}\) term is \(\binom{n}{6}\) and the coefficient of the \(13^{th}\) term is \(\binom{n}{12}\). These coefficients are given to be equal, i.e., \(\binom{n}{6} = \binom{n}{12}\).

Step 3: Property of Binomial Coefficients

One property of binomial coefficients is that \(\binom{n}{k} = \binom{n}{n-k}\). Using this property, equating \(\binom{n}{6} = \binom{n}{12}\) implies \(12 = n - 6\).

Step 4: Solve for n

From the equation \(12 = n - 6\), solve for \(n\) by adding \(6\) to both sides: \(n = 18\).

Key Concepts

Binomial coefficient

The binomial coefficient is a fundamental concept used to compute coefficients in the expansion of a binomial expression. It is represented with the notation \( \binom{n}{r} \), where \( n \) is the total number of items to choose from and \( r \) is the number of items being chosen. The formula for the binomial coefficient is:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

This formula counts the number of ways to choose \( r \) items from \( n \) items without considering the order of selection.
In the context of a problem involving the expansion of \((1+x)^n\), the binomial coefficient signifies the coefficient of a specific term in the expansion.

Binomial expansion

The binomial expansion refers to the process of expanding expressions raised to a power, like \((1+x)^n\). The expansion results in a series of terms, each with coefficients determined by the binomial coefficients. The general form of the binomial expansion is:

• \((1+x)^n = \sum_{r=0}^{n} \binom{n}{r} x^r\)

Each term is of the form \( \binom{n}{r} x^r \), where \( r \) represents the specific term's position, starting from zero.
The concept of binomial expansion is frequently used in algebra and probability theory to simplify expressions and to solve various problems involving powers of sums.

Mathematical properties

There are several important mathematical properties associated with binomial coefficients that simplify calculations and help solve problems. One such property is symmetry:

• \( \binom{n}{r} = \binom{n}{n-r} \)

This property states that choosing \( r \) elements from \( n \) elements is equivalent to choosing \( n-r \) elements, as the remaining elements from the total.
For the given problem, understanding symmetry helps deduce that equal coefficients at certain positions imply symmetry around the center of the expansion.
Another useful property is the relationship expressed with Pascal's identity:

• \( \binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1} \)

This expresses how each coefficient can be found as the sum of two coefficients from the previous level, facilitating easy calculation in recursive computations.