Problem 79 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(S=\sum_{r=0}^{m}{\underline{\phantom{xx}}}^{n+r} C_{k}\), then (A) \(S+{ }^{n} C_{k+1}={ }^{n+m} C_{k+1}\) (B) \(S+{ }^{n} C_{k+1}={ }^{n+m+1} C_{k+1}\) (C) \(S+{ }^{n} C_{k}={ }^{n+m} C_{k}\) (D) None of these

Step-by-Step Solution

Answer

(B) \(S+{ }^{n} C_{k+1}={ }^{n+m+1} C_{k+1}\) fits the solution.

Step 1: Understand the Task

We're given a sum involving binomial coefficients and need to determine its value in the context of the options provided. Specifically, we're looking for a match among the options for the sum \( S = \sum_{r=0}^{m} \binom{n+r}{k} \) plus another term.

Step 2: Analyze the Options

The exercise provides us with four different expressions involving binomial coefficients to compare with \( S + \binom{n}{k+1} \). Our goal is to simplify \( S \) and figure out which, if any, of these options it corresponds to.

Step 3: Apply Hockey Stick Identity

We use the Hockey Stick Identity (also known as the Christmas Stocking Theorem) which states: \[ \sum_{r=0}^{m} \binom{n+r}{k} = \binom{n+m+1}{k+1} - \binom{n}{k+1} \] Use this identity to express \( S \).

Step 4: Substitute and Simplify

Now replace the expression for \( S \) using the identity: \[ S = \binom{n+m+1}{k+1} - \binom{n}{k+1} \]Adding \( \binom{n}{k+1} \) to both sides gives: \[ S + \binom{n}{k+1} = \binom{n+m+1}{k+1} \]

Step 5: Compare with Options

Compare the result \( S + \binom{n}{k+1} = \binom{n+m+1}{k+1} \) with the listed options. Option B matches perfectly with our derived expression.

Key Concepts

Hockey Stick Identity

The Hockey Stick Identity, sometimes referred to as the Christmas Stocking Theorem, is a fascinating concept in combinatorics related to binomial coefficients. Imagine a "hockey stick" or "L" shape formed by numbers in Pascal's Triangle. The vertical handle of the stick consists of entries

• \( \binom{n}{k}, \binom{n+1}{k}, \ldots , \binom{n+m}{k} \)

Ending with the base

• \( \binom{n+m+1}{k+1} \)

The identity allows you to sum the entries in the handle (from \( n \) to \( n+m \)) and find that it equals the base minus the initial entry in the handle:\[\sum_{r=0}^{m} \binom{n+r}{k} = \binom{n+m+1}{k+1} - \binom{n}{k+1}.\]This powerful tool helps solve problems where you encounter successive binomial coefficients. In our exercise, leveraging this identity directly led us to simplifying and matching the expression to one of the options.

Binomial Theorem

The Binomial Theorem is a central tenet in algebra, used to expand expressions raised to a power. It states that for any integer \( n \) and real numbers \( a \) and \( b \),\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{k} b^{n-k}.\]This theorem shows how binomial coefficients are connected to the powers of two terms in a binomial expression. The coefficients themselves are gathered from Pascal's Triangle, making the theorem crucial for finding coefficients of expanded binomial terms.In combinatorial problems, the binomial theorem helps in calculating probabilities and combinations, as it lays the groundwork for understanding how different outcomes arise. For example, in determining how often a specific arrangement occurs in a sequence, you may use binomial coefficients derived directly from the theorem.

Combinatorics

Combinatorics is the study of counting, arrangement, and structure. It seeks to understand how elements interact under certain rules and constraints, making it vital for probability, data analysis, and testing scenarios.Under the realm of combinatorics, binomial coefficients often appear as solutions to counting problems. They represent the number of ways to choose \( k \) elements from a set of \( n \) elements, denoted by \( \binom{n}{k} \). This expression is also known as a "combination." Key aspects of combinatorics include:

• **Permutations:** Arrangements of objects in a particular order.
• **Combinations:** Selections of objects without regard to order.

The exercise focusing on binomial coefficients and the hockey stick identity is a pure combinatorial problem, where understanding the arrangement and counting of elements is essential. This branch of mathematics allows us to solve complex problems by organizing them into simpler tasks.