Problem 74 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(20 \%\) of three subsets (i.e., subsets containing exactly three elements) of the set \(A=\left\\{a_{1}, a_{2}, \ldots, a_{n}\right\\}\) contain \(a_{1}\), then the value of \(n\) is (A) 15 (B) 16 (C) 17 (C) 18

Step-by-Step Solution

Answer

The value of \(n\) is 15. Answer: (A) 15.

Step 1: Determine Total Number of 3-Element Subsets

The total number of 3-element subsets that can be selected from set \(A\) with \(n\) elements is given by the binomial coefficient \(\binom{n}{3}\). This represents all possible ways to pick 3 elements out of \(n\).

Step 2: Calculate Subsets Containing Element \(a_1\)

For subsets that contain the element \(a_1\), we need to select 2 additional elements from the remaining \(n-1\) elements. This can be done in \(\binom{n-1}{2}\) ways.

Step 3: Use Given Percentage Information

According to the problem, 20% of all 3-element subsets contain \(a_1\). Therefore, we set up the equation:\[\frac{\binom{n-1}{2}}{\binom{n}{3}} = 0.2\]

Step 4: Simplify Binomial Coefficients

Express the binomial coefficients in terms of factorials: \[\binom{n}{3} = \frac{n(n-1)(n-2)}{6}\]\[\binom{n-1}{2} = \frac{(n-1)(n-2)}{2}\]Substitute these into the equation:\[\frac{\frac{(n-1)(n-2)}{2}}{\frac{n(n-1)(n-2)}{6}} = 0.2\]

Step 5: Solve the Equation

Simplifying the equation, cancel \((n-1)(n-2)\) from numerator and denominator, then multiply both sides by 6:\[\frac{3}{n} = 0.2\]Multiply by \(n\) and solve for \(n\):\[3 = 0.2n\]\[n = \frac{3}{0.2} = 15\]

Step 6: Verify the Solution

Ensure the result satisfies the conditions by plugging \(n = 15\) back into original expressions. The number of 3-element subsets is \(\binom{15}{3}\), and subsets containing \(a_1\) is \(\binom{14}{2}\). Confirm their ratio is 0.2.

Key Concepts

Understanding Binomial Coefficients

Binomial coefficients are an essential part of combinatorics. They help us determine how many ways we can choose a certain number of elements from a larger set.
For example, if we want to select 3 elements from a set containing \(n\) elements, we use the binomial coefficient \(\binom{n}{3}\).
This coefficient is calculated using the formula:

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

Here, \(!\) represents a factorial, which is the product of all positive integers up to that number.
In our exercise, the binomial coefficient \(\binom{n}{3}\) tells us how many different 3-element subsets can be formed from set \(A\).
Understanding binomial coefficients simplifies tasks like finding the number of specific subsets containing certain elements.

Exploring Set Theory

Set theory is the foundation of understanding collections of objects. A set is a well-defined group of distinct objects, often referred to as elements.
In the given exercise, we deal with a set \(A\) containing \(n\) elements represented as \(\{a_1, a_2, \ldots, a_n\}\).
Sets can form subsets, which are smaller sets containing elements from the original set.
For instance, subsets of 3 elements from set \(A\) are of particular interest in our problem.

• They help us understand how mixing elements can create varied collections.
• The calculation of subsets contributes significantly to solving combinatorial problems.

By examining subsets, we can apply various mathematical tools, such as binomial coefficients, to explore properties and relationships within sets.
Each subset might have different criteria regarding which elements it includes, as explored in combinations containing specific set members like \(a_1\).

Approaching Mathematical Problem Solving

Mathematical problem solving involves various strategies to tackle challenges. In our exercise, the goal is to find a specific number, \(n\), based on given conditions.
To solve such problems:

• We start by comprehending the information provided, like percentages relating subset counts.
• We translate verbal conditions into mathematical equations, which in this case involve binomial coefficients.
• By expressing coefficients in terms of factorials, we simplify calculations.

In the given problem, we set up and simplify the equation:
\[\frac{\binom{n-1}{2}}{\binom{n}{3}} = 0.2\]We solve this equation step-by-step, reaching a solution by manipulating expressions, canceling terms, and verifying results.Verification is crucial. As we validate \(n = 15\) by recalculating subsets, we ensure that every aspect aligns with the problem's requirements.
Problem-solving doesn't only involve finding the answer.
It's also about understanding the process and applying reasoned logic to discover solutions that fit the given context.