Problem 66 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\left({ }^{\mathrm{m}} C_{0}+{ }^{\mathrm{m}} C_{1}-{ }^{\mathrm{m}} C_{2}-{ }^{\mathrm{m}} C_{3}\right)+\left({ }^{\mathrm{m}} C_{4}+{ }^{\mathrm{m}} C_{5}-{ }^{\mathrm{m}} C_{6}-{ }^{\text {" }} C_{7}\right)+\ldots\) \(=0\) if and only if for some positive integer \(k, m=\) (A) \(4 k\) (B) \(4 k+1\) (C) \(4 k-1\) (D) \(4 k+2\)

Step-by-Step Solution

Answer

The correct answer is (C) \(4k-1\).

Step 1: Understanding the Expression

The given expression is a series of combinations, or binomial coefficients, which are symmetric in nature. The terms are split into groups of four with alternating signs: \( \left({ }^{\mathrm{m}} C_{0}+{ }^{\mathrm{m}} C_{1}-{ }^{\mathrm{m}} C_{2}-{ }^{\mathrm{m}} C_{3}\right) \), and this pattern continues for subsequent terms.

Step 2: Analyze the Group Patterns

Each group has the structure: \( { }^{\mathrm{m}} C_{4k} + { }^{\mathrm{m}} C_{4k+1} - { }^{\mathrm{m}} C_{4k+2} - { }^{\mathrm{m}} C_{4k+3} \). Writing the next group, using patterns, is similar but offset by 4.

Step 3: Sum of Each Group

To find the sum of each group, notice that the binomial coefficients have a property that every group sums to zero for certain values of \(m\). Expanding one group:\( { }^{\mathrm{m}} C_{0} + { }^{\mathrm{m}} C_{1} - { }^{\mathrm{m}} C_{2} - { }^{\mathrm{m}} C_{3} \) simplifies to zero when \(m\) is of the form \(4k-1\).

Step 4: Check Symmetric Properties

Since we are given that the entire expression equals zero, it is crucial to find conditions where this symmetry repeats. When \(m\) is a multiple of 4 minus 1, that is \(m = 4k-1\), the symmetry leads to these sums cancelling out throughout.

Step 5: Finding the Correct Solution

Test the condition for each choice: (A) \(4k\), (B) \(4k+1\), (C) \(4k-1\), (D) \(4k+2\). Only \(4k-1\) gives the required property where each group of terms sums to zero.

Key Concepts

Binomial Coefficients

In mathematics, binomial coefficients are the numbers that appear in the expansion of a binomial raised to a power, written as \((a + b)^m\). These coefficients are represented as \( { }^m C_n \) or \( \binom{m}{n} \) and they have various properties that make them particularly useful.
One important property is that they actually count the number of ways to choose \(n\) items from \(m\) items, and hence they are quite central to combinatorics, as we will see further.
Binomial coefficients can be calculated using the formula:

• \( { }^m C_n = \frac{m!}{n!(m-n)!} \)

where \(!\) denotes the factorial of a number. This formula provides a method to find the coefficient for any given terms in the expansion of binomial expressions. Understanding how to compute these coefficients is crucial when analyzing algebraic expressions that involve a sequence of terms with alternating signs.

Combinatorics

Combinatorics is a field of mathematics focused on counting, arranging, and understanding the patterns within sets. It is directly related to binomial coefficients because these coefficients count the possible combinations of selecting items.
For our expression, which involves alternating sums and differences of binomial coefficients, combinatorial insights help discern how groupings contribute to overall symmetry.
The expression

• \(\left( { }^{m} C_{0} + { }^{m} C_{1} - { }^{m} C_{2} - { }^{m} C_{3} \right)\)

illustrates this combinatorial process by showing various ways elements can be selected contributing either positively or negatively to the final result. Such groupings are intentional to explore symmetry which can inform us when certain patterns result in zero sum outcomes. In combinatorics, recognizing these patterns is key to solving many similar problems.

Symmetric Properties

Symmetric properties are fundamental in understanding why certain combinations of binomial coefficients lead to results of zero. In this problem, by acknowledging that the expression is symmetric, we infer patterns that result in a repeated cancelation for specific values of \(m\).
When \(m = 4k-1\), the cumulative effect of the symmetrical properties is that all groups of coefficients sum to zero. This symmetry stems from how the coefficients are structured across the sequence.

• For example, each group is of the form \( { }^{m} C_{4k} + { }^{m} C_{4k+1} - { }^{m} C_{4k+2} - { }^{m} C_{4k+3} \)

Such symmetric properties reveal that altering the series by consistently grouped shifts leads to a zero sum under specific constraints, aligning perfectly when considering cyclic relationships within the coefficients. Understanding the role of these properties enables a simplified analysis of complex algebraic expressions.