Problem 47 - A Complete Resource Book in Mathematics for JEE Main 2017

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) \(S + { }^{n} C_{k+1} = { }^{n+m+1} C_{k+1}\)

Step 1: Understanding the Problem

We need to find the sum of the series \(S = \sum_{r=0}^{m} \binom{n+r}{k}\). We then need to verify equations given in the options by examining which aligns with this sum and manipulate it accordingly.

Step 2: Using the Hockey Stick Identity

The identity we will use is known as the "Hockey Stick" identity, which states: \( \sum_{r=0}^{m} \binom{n+r}{k} = \binom{n+m+1}{k+1} - \binom{n}{k+1} \). This identity simplifies cumulative binomial coefficients in a structured progression.

Step 3: Applying the Identity to S

By applying the Hockey Stick Identity, we have \( S = \sum_{r=0}^{m} \binom{n+r}{k} = \binom{n+m+1}{k+1} - \binom{n}{k+1} \). This represents the cumulative binomial sum from \(r=0\) to \(m\).

Step 4: Substituting in the Options

From \( S + \binom{n}{k+1} = \binom{n+m+1}{k+1} - \binom{n}{k+1} + \binom{n}{k+1} \). Simplifying gives us \( \binom{n+m+1}{k+1} \).

Step 5: Verification of Options

Compare \( S + \binom{n}{k+1} = \binom{n+m+1}{k+1} \) with the options:(A) \( S + \binom{n}{k+1} = \binom{n+m}{k+1} \) does not match.(B) \( S + \binom{n}{k+1} = \binom{n+m+1}{k+1} \) exactly matches our derived expression.(C) and (D) do not satisfy the equation either.

Key Concepts

Hockey Stick Identity

The "Hockey Stick Identity" is a fascinating and useful theorem in combinatorics that provides a way to sum specific binomial coefficients in a triangular pattern. Its name comes from the way the summed terms visually resemble a hockey stick when marked on Pascal's Triangle. This identity states:\[\sum_{r=0}^{m} \binom{n+r}{k} = \binom{n+m+1}{k+1} - \binom{n}{k+1}\]By summing the elements diagonally on Pascal's Triangle and then dropping down to the next row, the formula creates a perfect shape that connects terms quite effortlessly. This identity doesn't just simplify computations; it unravels patterns hidden within the triangle.
Pascal's Triangle serves as a helpful visualization tool when understanding this, providing a simple way to navigate the complex sums that would otherwise be cumbersome to handle individually.

Sum of Binomial Coefficients

The sum of binomial coefficients is a core component in combinatorics, featured prominently both in the Hockey Stick Identity and the Binomial Theorem. Binomial coefficients, denoted as \(\binom{n}{k}\), count the number of ways to choose \(k\) elements from a set of \(n\). Here, understanding how these coefficients accumulate is crucial to solving problems involving series and sums.
In our original exercise, we used the sum:

• \(S = \sum_{r=0}^{m} \binom{n+r}{k}\), where each term represents a binomial coefficient starting from \((n + 0)\) and ending at \((n + m)\).
• This creates a sequence of binomial coefficients that can be succinctly expressed using the Hockey Stick Identity.

Grasping this idea helps in simplifying complex series and finding elegant solutions to otherwise intricate expressions, demonstrating the beauty and utility of mathematical identities.

Binomial Theorem

The Binomial Theorem is a key mathematical principle that explains the expansion of binomials raised to a power. A binomial expression \(a + b\) raised to an integer power \(n\) can be expanded as:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Each term in this expansion involves a binomial coefficient \(\binom{n}{k}\), which indicates the number of possible ways to choose \(k\) items from \(n\) available items.
This theorem is fundamental not just in algebra but in probability and statistics, where it aids in defining distributions and solving problems related to combinatorial counting and polynomial expansions. The Binomial Theorem connects disparate areas of math, showcasing how recurring patterns manifest naturally across various topics, making it a cornerstone for further exploration into mathematical theories.