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

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: Understanding the Sum

The expression \( \binom{n}{m} + \binom{n-1}{m} + \binom{n-2}{m} + \cdots + \binom{m}{m} \) can be seen as a sum of combinations where the upper part of the binomial coefficient decreases while the lower part remains constant \( m \). This is essentially counting the number of ways to select items when varying the total number of available items.

Step 2: Recognizing the Pattern

Notice that the pattern formed by this sum is similar to the binomial expansion terms. The sum \( \sum_{k=m}^{n} \binom{k}{m} \) represents selecting \( m \) items from at most \( n \) items.

Step 3: Applying Identity

Using a combinatorial identity, we know that \( \sum_{k=m}^{n} \binom{k}{m} = \binom{n+1}{m+1} \). This identity checks all possible ways to choose \( m \) from sequences up to \( n \) plus another element, effectively considering an extra step not in the original range.

Step 4: Verifying Options

Now, compare \( \binom{n+1}{m+1} \) with the given options. (A) \( \binom{n+1}{m} \)(B) \( \binom{n+1}{m+1} \)(C) \( \binom{n}{m+1} \)(D) None of these. The identity matches precisely with option (B), \( \binom{n+1}{m+1} \).

Key Concepts

Combinatorial Identity

Combinatorial identities play a crucial role in solving problems related to combinations and permutations. These identities allow us to simplify complex expressions involving binomial coefficients and transform them into easier-to-evaluate forms. One such identity is the shifting identity applied in the exercise given:

• It states that the sum of binomial coefficients from a fixed lower index up to an upper index, i.e., \( \sum_{k=m}^{n} \binom{k}{m} \), can be expressed as \( \binom{n+1}{m+1} \) .

This identity helps us understand and compute the cumulative ways of choosing items, as it adds a hypothetical additional step in the selection process. By understanding this identity, we can easily verify the correct option given in the exercise, which is option (B).
Recognizing and utilizing such identities is essential for working efficiently with combinatorial problems.

Binomial Coefficient Sum

The summation of binomial coefficients involves computing the total number of ways to choose a specific number of items from a varying total. In the exercise, the sum \( \binom{n}{m} + \binom{n-1}{m} + \cdots + \binom{m}{m} \) is considered. Each term of this sum represents combinations where the lower term \( m \) remains constant while the upper term decreases.

• By summing these terms, we effectively account for every possible case where we can choose \( m \) elements from different total counts ranging from \( m \) to \( n \).

Understanding these sums involves recognizing the alternating choices as opportunities to include every possible scenario for selection. The overall result is a simplification into a single binomial term \( \binom{n+1}{m+1} \), according to the previously discussed identity. This simplification is what allows us to match the binomial coefficient sum with the correct choice (B) from the options provided in the exercise.

Binomial Expansion

Binomial expansion refers to expanding expressions of the form \((a+b)^n\) using binomial coefficients. Each expansion follows a fixed pattern rooted in Pascal's triangle, with each term taking the general form \( \binom{n}{k} a^{n-k}b^k \). This framework offers insights into how binomial coefficients behave under various algebraic manipulations.

• The exercise shows a conceptual similarity to binomial expansion. Although the exercise does not explicitly expand a binomial expression, the summation pattern is akin to selecting terms as it's done in binomial expansions.

This recognition is helpful for students as it aids in visualizing how choosing different combinations mirrors the visualization needed for expanding powers of binomials. Binomial coefficients, therefore, are not only tools for calculating combinations but also instrumental in expanding polynomial expressions efficiently.