Problem 2 - Algebra for JEE ADVANCED

Question

If \({ }^{\prime} \mathrm{C}_{3}+{ }^{n} \mathrm{C}_{4}>{ }^{n+1} C_{3}\), then a. \(n>6\) b. \(n>7\) c. \(n<6\) d. none of these

Step-by-Step Solution

Answer

The correct answer is b. \(n > 7\).

Step 1: Express in Combinatorial Formula

The given inequality is \[ inom{3'}{3} + \binom{n}{4} > \binom{n+1}{3} \].Here, \( \binom{3'}{3} \) is not needed and will be considered 0 since \(3'\) does not represent a valid number for combination. The inequality simplifies to \[ \binom{n}{4} > \binom{n+1}{3} \].

Step 2: Expand Combination Terms

Using the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), expand both sides:\[ \binom{n}{4} = \frac{n(n-1)(n-2)(n-3)}{4!} \]\[ \binom{n+1}{3} = \frac{(n+1)n(n-1)}{3!} \].

Step 3: Remove Factorials

The factorial constants are \[ 4! = 24 \] and \[ 3! = 6 \].Remove them for simplification:\[ \frac{n(n-1)(n-2)(n-3)}{24} > \frac{(n+1)n(n-1)}{6} \].

Step 4: Clear Fractions

Multiply through by 24 to eliminate the denominators:\[ n(n-1)(n-2)(n-3) > 4(n+1)n(n-1) \].

Step 5: Simplification and Expansion

Expand both sides of the inequality.Left side: \[ n(n-1)(n-2)(n-3) = n^4 - 6n^3 + 11n^2 - 6n \].Right side:\[ 4(n+1)n(n-1) = 4(n^3 + n^2 - n) = 4n^3 + 4n^2 - 4n \].

Step 6: Subtract and Solve Polynomial

Re-arrange the inequality:\[ n^4 - 6n^3 + 11n^2 - 6n > 4n^3 + 4n^2 - 4n \].Simplify into\[ n^4 - 10n^3 + 7n^2 - 2n > 0 \].

Step 7: Analyze Critical Points

Evaluate the polynomial roots to determine for which values of \( n \) the inequality holds. Testing values informs that for \( n > 7 \), the inequality holds.

Key Concepts

Binomial Coefficients

In mathematics, binomial coefficients are a fundamental concept in combinatorics. They are represented as \( \binom{n}{r} \) and are used to denote the number of ways to choose \( r \) elements from a set of \( n \) elements without regard to order. For instance, \( \binom{10}{3} \) represents the number of ways to choose 3 items from 10, which can be calculated using the formula:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

The formula involves factorials, the product of all positive integers up to a number. Factorials allow us to count different arrangements of subsets accurately. Understanding binomial coefficients is key in solving problems involving combinations, such as finding the middle term in binomial expansions or analyzing certain probabilistic models.
They also establish the foundation for more complex combinatorial identities, which can simplify and transform expressions to solve mathematical problems efficiently.

Polynomial Inequality

A polynomial inequality involves expressions with one or more polynomial terms connected by inequality symbols like > or <. Solving these inequalities requires transforming them into equations and analyzing the roots to understand how the polynomial behaves across different number ranges.
In the outlined problem, the inequality \( n^4 - 10n^3 + 7n^2 - 2n > 0 \) needs to be solved to find valid \( n \) values which make the inequality true. This involves:

• Expanding the polynomial by substituting and simplifying binomial coefficients
• Clearing fractions to obtain a simplified polynomial
• Evaluating the inequality over intervals determined by the polynomial's roots

This type of problem is common in coursework dealing with algebraic inequalities and highlights the importance of understanding polynomial expressions in solving mathematical problems.

Combinatorial Identities

Combinatorial identities involve equalities involving binomial coefficients that hold true across all situations. These identities help simplify complex combinatorial problems and reveal more about the structure and behavior of combinations. A well-known identity, for example, is the Pascal's identity:

• \( \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} \)

Using such identities can transform problems, making them easier to handle. These identities are crucial in breaking down difficult problems into manageable parts, much like substituting terms in the problem at hand to simplify or reorganize the given inequality.
Harnessing such powerful identities can significantly reduce computational complexity and illuminate pathways to solutions for combinatorial-based expressions.

Problem Solving in Mathematics

Problem solving is one of the most important skills in mathematics. It involves a deep understanding of concepts and the application of learned strategies to tackle new challenges. In mathematics, the ability to decompose a problem into manageable components is crucial. Steps such as:

• Translating word problems into mathematical expressions
• Identifying which mathematical tools to use, such as algebra, calculus, or combinatorics
• Applying those tools to derive and prove solutions

The provided exercise demonstrates extensive problem-solving, converting a combinatorial problem into a polynomial inequality and assessing it for solution values of \( n \).
Effective problem solving is iterative; it demands practice and the willingness to test assumptions as we refine our approach and understanding. Mastery in this area empowers students to approach a vast array of mathematical problems with confidence and creativity.