Problem 59 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The value of \(\sum_{r=0}^{20}{\underline{\phantom{xx}}}^{50-r} C_{6}\) is equal to: (a) \({ }^{51} C_{7}-{ }^{30} C_{7}\) (b) \({ }^{50} C_{7}-{ }^{30} C_{7}\) (c) \({ }^{50} C_{6}-{ }^{30} C_{6}\) (d) \({ }^{51} C_{7}^{\prime}+{ }^{30} C_{7}\)

Step-by-Step Solution

Answer

The value is (a) \( ^{51}C_7 - ^{30}C_7 \).

Step 1: Understand the Problem

We need to find the sum \( \sum_{r=0}^{20} \binom{50-r}{6} \). We recognize that each term is a binomial coefficient, which can be expanded and manipulated using properties of binomial coefficients.

Step 2: Use Symmetry Properties

Recall that \( \binom{n}{k} = \binom{n}{n-k} \) and that the summation can be expressed in terms of known identities involving binomial coefficients, such as the identity for sums of binomial coefficients.

Step 3: Apply Binomial Theorem Identity

Using the identity \( \sum_{k=0}^{n} \binom{n}{k} = 2^n \) and variant identities for coefficients, convert the given series to a recognizable form that makes calculation feasible.

Step 4: Simplify the Problem

We express the given sum \( \sum_{r=0}^{20} \binom{50-r}{6} \) as \( \sum_{k=0}^{20} \binom{50-k}{6} \), which translates to finding a cumulative count of specific outcomes in a larger set context.

Step 5: Calculate the Range Adjustment

Notice that \( \sum_{k=0}^{n} \binom{n}{k} \) gives the full set sum. Calculate the necessary subtraction to account for the upper bound cutoff by determining the excluded terms.

Step 6: Compute the Expression

Recognize that \( \sum_{k=0}^{50} \binom{50}{k} = \binom{51}{7} - \binom{30}{7} \). This switches the bounds and gets the result matching the given options.

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental principle in algebra that provides a formula for expanding binomials raised to any given power. A binomial is an expression containing two terms joined by a plus or minus sign, like \[(a + b)^n\].The theorem states:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

Each term in the expansion involves a binomial coefficient \(\binom{n}{k}\), which calculates the number of ways to choose \(k\) elements from a set of \(n\) elements. Binomial coefficients appear frequently in problems involving probability, algebra, and combinatorics.This theorem not only helps in expanding polynomial expressions but also in solving complicated summations involving these coefficients.

Summation Identities

Summation identities are powerful tools that simplify the calculation of sequences and series. In the context of binomial coefficients, there are several key identities:

• \( \sum_{k=0}^{n} \binom{n}{k} = 2^n \) - total possible subsets.
• \( \sum_{k=0}^{r} \binom{n}{k} = \binom{n+1}{r+1} \) - the hockey stick identity.

These identities leverage the properties of binomial coefficients to streamline computations. For example, the original exercise involves a summation of terms that are expressed as combinations of different upper indices. By recognizing these identities, you can transform them into more straightforward or known forms that are easier to evaluate.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects. It plays a crucial role in understanding the concept of binomial coefficients.In combinatorics, the binomial coefficient \(\binom{n}{k}\) is used to represent the number of ways to choose \(k\) items from \(n\) without regard to the order of selection. Some fundamental concepts in combinatorics include:

• Permutations: ordering of a set where order matters.
• Combinations: selecting from a set where order does not matter.
• Factorials: used in calculating permutations and combinations.

Understanding these combinatorial principles is vital for tackling problems related to arrangements, groupings, and probability.

Symmetry Properties

Symmetry properties in binomial coefficients showcase their inherent balance and simplicity in mathematical expressions. An essential property is:

• \(\binom{n}{k} = \binom{n}{n-k}\)

This reflects the idea that choosing \(k\) items from \(n\) is equivalent to leaving \(n-k\) items behind. This property can simplify the computation and understanding of binomial coefficients, especially in complex sums and algebraic manipulations.Another significant symmetry property involves Pascal's triangle, where each number is the sum of the two numbers directly above it, illustrating a natural symmetry in binomial distribution. Understanding and using these symmetry properties can greatly assist in simplifying and resolving binomial-related problems.