Problem 38 - A Complete Resource Book in Mathematics for JEE Main 2017

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.

Step 1: Understand the Total Number of Subsets

The total number of subsets containing exactly three elements from a set with \( n \) elements is sometimes denoted by \( \binom{n}{3} \). This is a combination formula representing how many ways we can choose 3 elements from \( n \).

Step 2: Calculate Subsets Containing \( a_1 \)

For a subset to contain \( a_1 \), we need to choose the remaining 2 elements from the other \( n-1 \) elements. So, the number of subsets containing \( a_1 \) is \( \binom{n-1}{2} \).

Step 3: Set Up the Equation Using 20% Condition

Given that 20% of all subsets contain \( a_1 \), we can set up the equation: \[ \frac{\binom{n-1}{2}}{\binom{n}{3}} = 0.2 \].

Step 4: Determine \( \binom{n}{3} \) and \( \binom{n-1}{2} \)

Calculate \( \binom{n}{3} = \frac{n(n-1)(n-2)}{6} \) and \( \binom{n-1}{2} = \frac{(n-1)(n-2)}{2} \), which will be used to set the equation.

Step 5: Simplify the Equation

Substitute \( \binom{n-1}{2} = \frac{(n-1)(n-2)}{2} \) and \( \binom{n}{3}= \frac{n(n-1)(n-2)}{6} \) into the equation: \[ \frac{(n-1)(n-2)/2}{n(n-1)(n-2)/6} = 0.2 \].

Step 6: Solve the Equation for \( n \)

Simplify the left side of the equation: \( \frac{3}{n} = 0.2 \). Solve for \( n \) by cross-multiplying: \( 3 = 0.2n \). Hence, \( n = \frac{3}{0.2} = 15 \).

Step 7: Verify the Solution

Verify each step to ensure calculations are consistent with the original equation. The calculated \( n \), after simplification, satisfies all given conditions.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. It involves the study of finite or countable discrete structures. In the context of subset problems, combinatorics helps us to determine how many different ways we can select a subset from a set.
To solve such problems, we use concepts like permutations and combinations.
A permutation is an arrangement of objects in a specific order, while a combination is a selection of objects without regard to order.

• In the given exercise, we are dealing with combinations because order doesn't matter when choosing subsets.
• Therefore, understanding combinatorics is key to counting the number of subsets from a set.

Binomial Coefficient

The binomial coefficient, often represented as \( \binom{n}{k} \), plays a crucial role in combinatorial calculations.
It represents the number of ways to choose \( k \) elements from a set of \( n \) elements. The formula is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]This equation is fundamental in problems involving subset selections.

• For example, in the exercise, \( \binom{n}{3} \) is used to determine how many subsets of exactly three elements can be formed from a set of \( n \) elements.
• Another key point is understanding how to calculate the number of subsets that specifically include a given element, such as \( a_1 \).
• This is done using \( \binom{n-1}{2} \), selecting 2 more elements from the remaining \( n-1 \) elements.

Thus, the binomial coefficient helps simplify determining the exact number of desired subsets.

Mathematical Equations

Mathematical equations are used to represent relationships between different quantities. In our exercise, equations help us link the number of subsets containing a specific element to the total number of possible subsets.
When the problem states that 20% of these subsets contain \( a_1 \), we express this relationship using an equation:\[\frac{\binom{n-1}{2}}{\binom{n}{3}} = 0.2\]This equation allows us to mathematically enforce the condition that 20% of the subsets have a certain property.

• The numerator \( \binom{n-1}{2} \) represents subsets containing \( a_1 \).
• The denominator \( \binom{n}{3} \) represents the total number of subsets with three elements.
• Simplifying and solving these equations yields the value of \( n \) which meets all conditions of the problem.

Set Theory

Set theory is a mathematical theory that studies collections of objects, known as sets. This theory provides the foundation for understanding concepts such as subsets, intersections, and unions. In our exercise, we need to comprehend how subsets work:

• A subset is simply a set composed of elements from another set.
  Every set can have several subsets, including the empty set and the set itself.
• In the exercise, we are interested in subsets made up of exactly three elements. Understanding how to count and manipulate these subsets is crucial.
• We specifically need to focus on subsets that include a particular element, say \( a_1 \).

Using these fundamental principles of set theory helps us gain a deeper understanding of the problem and guide us in forming solutions.