Problem 47 - 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. \(S + \binom{n}{k+1} = \binom{n+m+1}{k+1}\).

Step 1: Recognize the Summation

The given sum is \(S = \sum_{r=0}^{m} \binom{n+r}{k}\). We need to find which equation satisfies this sum.

Step 2: Use Identity for Summation of Combinations

A useful identity here is \(\sum_{r=0}^{m} \binom{n+r}{k} = \binom{n+m+1}{k+1} - \binom{n}{k+1}\), which comes from the hockey-stick identity for combinations.

Step 3: Add \(\binom{n}{k+1}\) to Both Sides

To solve for \(S + \binom{n}{k+1}\), add \(\binom{n}{k+1}\) to the right-hand side of the identity: \(\sum_{r=0}^{m} \binom{n+r}{k} + \binom{n}{k+1} = \binom{n+m+1}{k+1}\).

Step 4: Identify the Correct Option

The expression now is \(S + \binom{n}{k+1} = \binom{n+m+1}{k+1}\). This matches with option B in the provided list of options.

Key Concepts

binomial coefficients

One of the cornerstones of combinatorics, binomial coefficients, are numbers that describe the number of ways to choose a subset of items from a larger set. This is typically represented as \( \binom{n}{k} \), which is read as "n choose k". This denotes the number of ways to choose \( k \) elements from a set of \( n \) elements, disregarding the order of selection.

The formula to compute a binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( ! \) denotes factorial, the product of all positive integers up to that number. Binomial coefficients are useful for solving problems related to combinations and permutations, as they provide a quick way to count combinations.

• \( \binom{5}{2} = \frac{5!}{2!(5-2)!} = 10 \)
• \( \binom{10}{3} = \frac{10!}{3!(10-3)!} = 120 \)

Binomial coefficients appear in the expansion of binomial expressions, like in the binomial theorem: \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). This connects them to algebra and analysis, making them versatile in solving various mathematical problems.

hockey-stick identity

The hockey-stick identity is a delightful result in combinatorics related to binomial coefficients. It visually resembles a hockey stick when plotted, which is how it gets its name. This identity provides a neat way to sum specific sequences of binomial coefficients.

Mathematically, the hockey-stick identity states that:
\[\sum_{r=0}^{m} \binom{n+r}{k} = \binom{n+m+1}{k+1} - \binom{n}{k+1}. \]
This is incredibly useful for computing the sum of combinations in a straightforward manner, simplifying calculations that might otherwise seem complex.

• Imagine a vertical column of binomial coefficients starting at \( \binom{n}{k+1} \).
• Then there is a diagonal running from \( \binom{n}{k+1} \) upwards to \( \binom{n+m}{k+1} \).

The sum of this "hockey stick" of binomial coefficients helps to quickly find the desired results. This identity is often utilized when the sequence aligns perfectly with the hockey-stick pattern, allowing significant simplification of the problem at hand.

summation of combinations

Summation of combinations is a technique in combinatorics used to handle problems where you need to add together a series of binomial coefficients. This approach is very useful for breaking down complex combinatorial problems into manageable parts.

In essence, if you have a sum like \( \sum_{r=0}^{m} \binom{n+r}{k} \), applying identities like the hockey-stick identity can simplify the calculation. The benefit of summing combinations is that it converts what might appear as a complicated problem into a simpler arithmetic one, leveraging known patterns and shortcuts.

• It allows for the simplification of nested or repeated summations.
• Helps in identifying patterns or relationships between different combinations.

Beyond computational usefulness, understanding the summation of combinations enriches one's grasp of the symmetrical properties of combinatorial numbers, helping solve a wide range of theoretical and applied mathematics problems.