Problem 9 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a\) represents the number of permutations of \((x+2)\) things taken together, \(b\) represents the number of permutations of 11 things taken together out of \(x\) things, and \(c\) represents the number of permutations of \((x-11)\) things taken together so that \(a=182 b c\), then \(x=\) (A) 15 (B) 12 (C) 10 (D) 18

Step-by-Step Solution

Answer

The value of \( x \) is 12.

Step 1: Understanding the Problem

We need to express the given conditions in mathematical form. We are given that \( a \) represents the number of permutations of \( (x + 2) \) things, \( b \) represents the number of permutations of 11 things from \( x \) things, and \( c \) represents the permutations of \( (x-11) \) things. We also know that \( a = 182b \times c \). Our goal is to find the value of \( x \).

Step 2: Expressing Permutations

The permutation of \( n \) items is given by \( n! \). Therefore:- \( a = (x + 2)! \)- \( b = \frac{x!}{(x - 11)!} \)- \( c = (x - 11)! \)

Step 3: Substitute and Simplify

Substitute the expressions for \( a \), \( b \), and \( c \) into the equation:\[(x+2)! = 182 \times \frac{x!}{(x-11)!} \times (x-11)!\]This simplifies to: \[(x+2)! = 182 \times x!\]

Step 4: Simplify Further

We can cancel \( x! \) from both sides:\[(x+2) \times (x+1) = 182\]Now solve for \( x \).

Step 5: Solve for x

Expand and rearrange:\[x^2 + 3x + 2 = 182\]Which leads to:\[x^2 + 3x - 180 = 0\]Now, we solve this quadratic equation.

Step 6: Quadratic Solution

We solve \( x^2 + 3x - 180 = 0 \) using the quadratic formula: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]Here, \( a = 1, b = 3, c = -180 \). Calculate the discriminant:\[ \sqrt{{3^2 + 720}} = \sqrt{729} = 27 \]Now solve:\[x = \frac{{-3 \pm 27}}{2}\]This gives the solutions \( x = 12 \) and \( x = -15 \), but since \( x \) must be positive, \( x = 12 \).

Key Concepts

Factorials

Factorials provide us with a way to determine the number of permutations by calculating the total possible arrangements of a set of objects. In simple words, if you have a set of objects and want to know how many different orders you can arrange them in, you use a factorial to find out.
The factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). Let's take a closer look with examples:

• For \( 3! \): Calculate \( 3 \times 2 \times 1 = 6 \)
• For \( 5! \): Calculate \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Factorials are key in calculating permutations. They help determine how items can be arranged in every possible sequence, which is precisely what's needed when solving permutation problems.

Quadratic Equations

Quadratic equations are a type of polynomial equation with a degree of two. They are generally written in the standard form \( ax^2 + bx + c = 0 \). Quadratic equations are fundamental in various calculations in mathematics and science.
The key methods to solve quadratic equations include:

• Factoring: Writing the quadratic as a product of two binomials.
• Completing the square: Rewriting the quadratic into a squared binomial form.
• Quadratic formula: Using \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots of the equation.

In the provided exercise, \( x^2 + 3x - 180 = 0 \) is the quadratic equation used. By applying the quadratic formula, we identify the possible values for \( x \), finding \( x = 12 \) as the valid solution. The discriminant, \( b^2 - 4ac \), determined it is \( 27 \), giving us solutions with practical application in permutations.

Mathematical Problem Solving

Mathematical problem solving involves a logical sequence of steps that ensue from understanding the problem to arriving at a solution. It is a skill crucial for tackling complex issues that require clear strategies and thoughtful analysis.
Here's a breakdown of effective problem-solving steps:

• Understand the Problem: Read the problem carefully and identify what is being asked.
• Devise a Plan: Determine how you will approach solving the problem. This might include selecting formulas or mathematical operations needed.
• Carry Out the Plan: Implement the steps from your plan computationally, strictly following mathematical logic.
• Review and Recheck: Ensure the solution makes sense in context to the problem initially posed.

This approach was used to solve the exercise where permutations were equated using factorial terms and simplified to find the unknown value. Importantly, a consistent problem-solving approach helps ensure accuracy and leads to correct solutions in mathematics.