Problem 17 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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

Step-by-Step Solution

Answer

(B) \({ }^{n+1} C_{r+1}\)

Step 1: Understanding the Problem

We're given an expression where the left-hand side is the sum of binomial coefficients from \(^{m} C_{r}\) to \(^{n} C_{r}\). We need to identify what this sum equals by comparing it with the provided choices.

Step 2: Using Binomial Theorem

The binomial theorem states that the sum of binomial coefficients along a horizontal line of Pascal's triangle can be expressed using a specific relationship: \(\sum_{k=r}^{n} { }^{k} C_{r} = { }^{n+1} C_{r+1}\). We will apply this theorem to our given sum.

Step 3: Applying the Formula

Substitute \(r\) with \(r+1\) in the binomial theorem formula: \(\sum_{k=m}^{n} { }^{k} C_{r} = { }^{n+1} C_{r+1}\). By applying this to our problem, the expression \(\sum_{k=m}^{n} { }^{k} C_{r+1}\) should simplify to \({ }^{n+1} C_{r+1}\).

Step 4: Comparing with Options

The simplified expression \({ }^{n+1} C_{r+1}\) matches option (B) among the given choices. Thus, the correct answer to the exercise is option (B).

Key Concepts

Pascal's Triangle

Let's begin with Pascal's Triangle, a fascinating tool in mathematics named after the French mathematician Blaise Pascal. It's a triangular array of numbers where each number is the sum of the two numbers directly above it. This structure beautifully displays the properties of binomial coefficients.

• The first row is simply \(1\).
• Each subsequent row starts and ends with \(1\), and each interior number is the sum of the two numbers above it.

For example, the third row, apart from the leading and trailing \(1\), contains \(2\) because \(1+1=2\).

Pascal's Triangle is used extensively in probability and combinatorics because it represents the coefficients in the binomial expansion \((a + b)^n\). It provides a quick and easy way to determine these coefficients, making it an essential tool in solving problems related to the Binomial Theorem.

Binomial Coefficients

Binomial coefficients are the numbers that appear in the expansion of a binomial raised to a power, such as \((a+b)^n\). They are denoted as \({ }^{n}C_{k}\) or \( \binom{n}{k} \), where \(n\) is the power to which the binomial is raised, and \(k\) indicates the specific term in the expansion.

• The binomial coefficient \({ }^{n}C_{k}\) is calculated using the formula: \( \frac{n!}{k!(n-k)!} \).
• Here, \(!\) denotes a factorial, which is the product of all positive integers up to that number.

These coefficients are critical in finding probabilities and combinations because they count the number of ways to choose \(k\) objects from \(n\) without regard to order. This makes them incredibly useful in fields such as combinatorics and statistics, and they conveniently align with the rows of Pascal's Triangle.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. It's a vast field that encompasses various methods for quantifying the likelihood of particular outcomes or configurations.

A few key concepts in combinatorics include:

• Permutations - Arrangements of objects where order matters.
• Combinations - Selection of objects where order does not matter. Binomial coefficients are often used here to determine the number of combinations.
• Pascal's Triangle - As discussed, it is a powerful tool in combinatorics for finding binomial coefficients and solving related problems.

In the context of the given exercise, combinatorics helps us work out the sum relationships between coefficients using the Binomial Theorem. Understanding combinatorics opens doors to solving complex problems not only in mathematics but also in computer science, probability theory, and algorithm design.