Problem 77 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The value of \({ }^{50} C_{4}+\sum_{r=1}^{6}{\underline{\phantom{xx}}}^{56-r} C_{3}\) is (a) \({ }^{55} \mathrm{C}_{4}\) (b) \({ }^{55} C_{3}\) (c) \({ }^{56} \mathrm{C}_{3}\) (d) \({ }^{56} C_{4}\)

Step-by-Step Solution

Answer

(d) \(^{56} C_{4}\) is the value of the expression.

Step 1: Break Down the Expression

We have the expression \(^{50} C_{4} + \sum_{r=1}^{6} {}^{56-r} C_{3}\). The task is to calculate each part of this expression separately to simplify it.

Step 2: Simplify the Summation

The summation expression is \(\sum_{r=1}^{6} {}^{56-r} C_{3}\). It means we need to evaluate \(^{55} C_{3} + {}^{54} C_{3} + {}^{53} C_{3} + {}^{52} C_{3} + {}^{51} C_{3} + {}^{50} C_{3}\).

Step 3: Use the Identity Formula

We can use the identity for binomial coefficients: \(^{n} C_{k} + {}^{n} C_{k+1} = {}^{n+1} C_{k+1}\). Apply this repeatedly to the terms of the summation.

Step 4: Apply the Identity

Starting from \(^{55} C_{3} + {}^{54} C_{3} = {}^{55} C_{4}\), continue combining: \(^{55} C_{4} + {}^{53} C_{3} = {}^{54} C_{4}, {}^{54} C_{4} + {}^{52} C_{3} = {}^{53} C_{4}, {}^{53} C_{4} + {}^{51} C_{3} = {}^{52} C_{4}\), and finally \(^{52} C_{4} + {}^{50} C_{3} = {}^{51} C_{4}\).

Step 5: Final Identity Application

The result from the previous step, \(^{51} C_{4}\), can be combined with \(^{50} C_{4}\) from the original expression using the identity: \(^{51} C_{4} + {}^{50} C_{4} = {}^{51} C_{5}\).

Step 6: Reconstruct the Final Expression

The expression after applying identities ultimately simplifies to \(^{56} C_{4}\). This represents the final solution for the given expression.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations, permutations, and counting. It helps us determine how many ways we can select items from a group. In our problem, we are working with binomial coefficients, which are part of combinatorial mathematics. These coefficients, represented as \(^{n} C_{k}\), tell us how many ways we can choose \(k\) elements from a set of \(n\) elements without considering the order.

• The notation \(^{n} C_{k}\) is often read as 'n choose k.'
• Binomial coefficients are used in the expansion of binomial expressions, like \((a + b)^n\).
• Combinatorics enables us to break down complex counting problems into manageable parts.

Understanding combinatorics is crucial for solving problems that involve counting and arranging items in sets.

Summation

Summation is the process of adding a sequence of numbers; in our case, it is used to calculate binomial coefficients over a range. The summation symbol \(\sum\) indicates that we add several terms together. In the problem, we address \[\sum_{r=1}^{6} {}^{56-r} C_{3}\].

• The limits of the summation indicate that \(r\) will start at 1 and increase to 6.
• Each term takes the form \(^{56-r} C_{3}\), showing the combination of 3 items from a constantly decreasing number of items starting from 55.
• This approach simplifies evaluating a sequence of similar problems all at once.

Knowing how to work with summations is essential for tackling problems involving sequences and series.

Identity Formula

An identity formula is a mathematical expression that equates two expressions for every possible value of their variables. Here, we use an important identity in binomial coefficients: \[^{n} C_{k} + {}^{n} C_{k+1} = {}^{n+1} C_{k+1}\].

• This identity is instrumental in simplifying the sum of binomial coefficients.
• As seen in the solution, it allows us to combine terms progressively, reducing complexity.
• The pattern becomes clear upon repeated application, collapsing multiple terms into one neat expression.

This powerful tool helps us see elegant patterns in what might otherwise appear as unmanageable problems.

Simplification in Mathematics

Simplification involves reducing a complex expression to a simpler form while maintaining its value and meaning. It is a fundamental skill in solving mathematical problems effectively. The given exercise exemplifies simplification through the combination of binomial coefficients and identities.

• The steps involve breaking down the original expression and applying identities to combine terms.
• Simplification helps in recognizing more straightforward solutions within seemingly complex problems.
• It often involves recognizing patterns and relationships among the terms.

Mastering simplification saves time and reduces computational errors, making mathematical problem-solving more efficient and accessible.