Problem 105 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

For any positive integers \(m, n\) (with \(n \geq m\) ), let and \(\left(\begin{array}{l}n \\ m\end{array}\right)={ }^{n} C_{m}\), then \(\left(\begin{array}{l}n \\ m\end{array}\right)+\left(\begin{array}{l}n-1 \\\ m\end{array}\right)+\left(\begin{array}{l}n-2 \\\ m\end{array}\right)+\ldots+\left(\begin{array}{c}m \\ m\end{array}\right)=\) (A) \(\left(\begin{array}{l}n+1 \\ m\end{array}\right)\) (B) \(\left(\begin{array}{l}n+1 \\ m+1\end{array}\right)\) (C) \(\left(\begin{array}{c}n \\ m+1\end{array}\right)\) (D) None of these

Step-by-Step Solution

Answer

The answer is (B) \( \binom{n+1}{m+1} \).

Step 1: Understand the Problem

We need to determine the sum of a series of binomial coefficients: \( \binom{n}{m} + \binom{n-1}{m} + \binom{n-2}{m} + \ldots + \binom{m}{m} \). We will use properties of binomial coefficients and combinatorial identities to find this sum.

Step 2: Recognize the Pattern

The sequence given: \( \binom{n}{m}, \binom{n-1}{m}, \ldots, \binom{m}{m} \) resembles parts of a binomial coefficient identity, specifically the identity used for downward sums. Consider a possible identity that helps with sums like \( \binom{n}{k} + \binom{n-1}{k} + \cdots + \binom{k}{k} \).

Step 3: Apply the Relevant Identity

There's a relevant identity in combinatorics: \( \binom{n}{k} + \binom{n-1}{k} + \cdots + \binom{k}{k} = \binom{n+1}{k+1} \). This translates here to: \( \binom{n}{m} + \binom{n-1}{m} + \cdots + \binom{m}{m} = \binom{n+1}{m+1} \).

Step 4: Compare with Problem Options

Recall the choices provided. The matching expression from the identity is \( \binom{n+1}{m+1} \), which corresponds to option (B).

Step 5: Conclusion and Final Answer

Using the binomial coefficient identity, the sum of the given sequence is \( \binom{n+1}{m+1} \), which matches option (B). Therefore, the correct choice is (B) \( \binom{n+1}{m+1} \).

Key Concepts

Combinatorial Identities

Combinatorial identities play a crucial role in simplifying and solving problems involving binomial coefficients. These involve understanding how combinations of objects can be added, removed, or rearranged in ways that can be described with an equation.
Consider, for instance, a simple identity: \( \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} \). This identity reflects that any group of \(k\) items chosen from \(n\) items can either include a specific item or not, leading to an addition of two distinct groups of choices.
In these identities, each term in a sequence can represent a way to count the number of solutions to a problem. Combinatorial identities help verify or discover new ways to count by establishing equality through known expressions.

• They help in connecting different combinatorial expressions.
• They can provide elegant proofs to complex problems.
• They are essential for developing new mathematical formulas.

Understanding these identities simplifies calculations and helps in recognizing patterns and relationships that might not otherwise be obvious.

Pascal's Identity

Pascal's Identity is a fundamental combinatorial identity that is simple but very powerful. It is named after the French mathematician Blaise Pascal.
This identity is expressed as: \[\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} \] What this means is if you want to choose \(k\) items from \(n\) items, you have two choices for the first item: include it or not. The identity breaks down this choice into :

• \(\binom{n-1}{k-1}\): choosing the first item and then \(k-1\) items from the remaining \(n-1\).
• \(\binom{n-1}{k}\): not choosing the first item and picking all \(k\) items from the rest \(n-1\) items.

Pascal's Identity is visually easy to understand through Pascal's Triangle, where each number is the sum of the two numbers directly above it.
This identity holds up across different levels of problems and illustrates a basic structure of recursive computation. It's not only a useful computational tool but also foundational in understanding more complex identities.

Mathematical Induction

Mathematical induction is a key proof technique used widely in mathematics. It is especially useful for proving statements about integers. This method works like a domino effect, where knocking over the first domino causes the whole series to fall.
The process of induction involves two main steps:

• Base Case: Prove the statement is true for the initial value.
• Inductive Step: Show if the statement is true for an arbitrary value \(n\), then it is also true for \(n+1\).

Applying these steps effectively proves the statement for all natural numbers.
Using induction with binomial coefficients often involves proving identities or inequalities. Induction helps ensure the structural reliability of these identities over potentially infinite cases.
Teaching induction through examples, such as proving combinatorial identities, can deeply enhance the understanding of this concept. This makes mathematical induction a cornerstone of logical reasoning in mathematics.