Problem 22 - A Complete Resource Book in Mathematics for JEE Main

Question

These are four balls of different colours and four boxes of colours, same as those of the balls. The number of ways in which the balls, one each in a box, could be placed such that a ball does not go to a box of its own colour is (A) \(\frac{5}{8}\) (B) \(\frac{3}{8}\) (C) \(\frac{1}{8}\) (D) none of these

Step-by-Step Solution

Answer

There are 9 ways; correct answer is (D) none of these.

Step 1: Understand Derangements

A derangement is a permutation where no object appears in its original position. In this problem, it means no ball goes into a box of its matching color.

Step 2: Calculate Derangement for Four Items

The formula for calculating the number of derangements of n items is \[ !n = n! \sum_{i=0}^{n} \frac{(-1)^i}{i!} \]For four items: \[ !4 = 4! \left(\frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!}\right) \] Calculate each term of the sum: \[ 4! = 24 \] Thus:\[ !4 = 24 \left(1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24}\right) \]

Step 3: Simplify the Expression

Calculate step by step: - \(1 - 1 = 0\)- Adding \(\frac{1}{2}\) gives \(0 + \frac{1}{2} = \frac{1}{2}\)- Subtracting \(\frac{1}{6}\) gives \(\frac{1}{2} - \frac{1}{6} = \frac{3}{6} - \frac{1}{6} = \frac{2}{6} = \frac{1}{3}\)- Adding \(\frac{1}{24}\) gives \(\frac{1}{3} + \frac{1}{24} = \frac{8}{24} + \frac{1}{24} = \frac{9}{24} = \frac{3}{8}\)

Step 4: Multiply the Simplified Expression by 24

Finally, multiply the simplified expression by 24: \[ !4 = 24 \times \frac{3}{8} = 24 \times 0.375 = 9 \]Thus, there are 9 ways to arrange the balls such that none goes into a box of its own color.

Step 5: Conclusion

The result of 9 ways does not correspond to any of the given answer choices. Therefore, the correct answer is (D) none of these.

Key Concepts

Permutations

A permutation is a specific arrangement of a set of objects. Imagine you have a group of four differently colored balls, and you want to organize them in a line. Each distinct order represents a unique permutation. Permutations are vital because they detail how things can be arranged in different ways.
For a given number of objects, say \( n \), the number of possible permutations is depicted by factorial notation, \( n! \). This notation represents the total permutations possible for arranging \( n \) distinct items into a sequence. In mathematical terms, if you have a set of \( n \) objects, the number of possible distinct arrangements is \( n! \). Each element has a unique spot in the sequence, much like placing the balls in distinct boxes without regard to color.
Understanding permutations helps in solving problems like derangements, where you need specific arrangements that follow certain restrictions.

Combinatorics

Combinatorics is the mathematical study of counting, arrangements, and combinations. It involves understanding how sets of items can be mixed and organized in various ways. By using combinatorial methods, we can solve problems related to selection and arrangement.
In the problem of derangements, combinatorics helps us figure out how balls can be placed in boxes such that no ball ends up in a color-matching box. The concept applies advanced combinatorial formulas to account for this unique arrangement requirement.

• Counting principles are employed to determine possible arrangements.
• Factorials are used in calculations to represent the total number of ways items can be organized.
• The principle of inclusion-exclusion often helps calculate derangements by including favorable outcomes and excluding unfavorable ones.

Combinatorial formulas create frameworks to solve these type of questions efficiently, blending parts of permutations and probability.

Factorials

Factorials are a mathematical tool expressed by an exclamation mark, like \( n! \). They indicate the result of multiplying a sequence of descending natural numbers. For instance, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \), representing all the different ways four items can be arranged in sequence.
Factorials simplify the way we count permutations. Consider how they quickly allow us to calculate total arrangements without listing each possibility. This is particularly useful in larger data sets, where manually counting arrangements is impractical. Factorials are integral to many combinatorial formulas, including the calculation of derangements. When handling questions like the one about colored balls and boxes, understanding factorials allows for calculation of possible arrangements quickly, especially when applying formulas such as the derangement equation.
By providing a basis for counting permutations and combinations, factorials help navigate through complex arrangement problems by offering a concise and definitive way to determine total possibilities.