Problem 84 - A Complete Resource Book in Mathematics for JEE Main

Question

There 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

The number of ways is 9, which matches option (D) 'none of these.'

Step 1: Understand the Problem

We need to arrange 4 balls into 4 boxes of the same colors such that no ball is placed into a box with the same color. This is a classic example of a problem involving permutations without fixed points, also known as derangements.

Step 2: Define the Derangement Formula

To find the number of derangements (a permutation where no element appears in its original position) for 4 items, we use the formula for the number of derangements (\(!n\)): \(!n = n! \sum_{i=0}^{n} \frac{(-1)^i}{i!}\).

Step 3: Calculate the Factorial of 4

Calculate \(4!\), which is the factorial of the number of balls/boxes: \(4! = 4 \times 3 \times 2 \times 1 = 24\).

Step 4: Compute the Sum for the Derangement Formula

Compute the sum \(\sum_{i=0}^{4} \frac{(-1)^i}{i!}\). This expansion yields: \(\frac{(-1)^0}{0!} + \frac{(-1)^1}{1!} + \frac{(-1)^2}{2!} + \frac{(-1)^3}{3!} + \frac{(-1)^4}{4!} = 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24}\).

Step 5: Simplify the Expression for the Sum

Simplify the expression: \(1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} = 0 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24}\). Simplifying further: \(\frac{1}{2} = \frac{12}{24}, \frac{1}{6} = \frac{4}{24}, \frac{1}{24} = \frac{1}{24}\). Thus: \(\frac{12}{24} - \frac{4}{24} + \frac{1}{24} = \frac{9}{24}\).

Step 6: Calculate the Deranged Arrangements

Using the formula \(!n = n! \times \text{sum from Step 5}\), compute \(!4 = 24 \times \frac{9}{24} = 9\). This means there are 9 ways to arrange the balls such that no ball is in a box of the same color.

Step 7: Verify with the Given Options

None of the given options directly represent the number of ways calculated. Verify whether any option is a reduced fraction of another form of these results. Conclusion: none of the options (A), (B), and (C) represent 9 as a fraction of the total possible arrangements, because for 4 distinct arrangements, the fraction should have resulted in fractions in terms of precise parts/ratios of these arrangements and not matching this exact concrete number.

Key Concepts

Permutations

In mathematics, permutations refer to the different ways in which a set of distinct objects can be arranged or ordered. When arranging 4 balls into 4 boxes, we consider the concept of permutations because we want to understand all possible configurations of these 4 objects.
In a permutation, order matters, making it distinct from combinations. For instance, the arrangement ABCD is different from BACD. When we speak of permutations, especially in the context of derangements, we're interested in specific subsets of these arrangements. That is, arrangements where none of the items end up in their original positions, which leads us to derangements. This special type of permutation is critical in solving the exercise presented.
This understanding of permutations allows us to delve deeper into arranging objects in discrete mathematics, especially crafting unique orders under specific set conditions.

Factorials

Factorials, denoted by an exclamation mark (e.g., 4!), play a crucial role in understanding permutations and combinatorics. A factorial represents the product of an integer and all the non-negative integers below it. Thus, 4! is calculated as:
\[4! = 4 \times 3 \times 2 \times 1 = 24\]
Factorials are essential in computing permutations because they help in counting the total number of possible arrangements of a set of objects. In the case of the derangement problem, calculating the factorial of 4 provides the total arrangements before applying the derangement formula.
By mastering factorials, one can efficiently tackle quickly how large groups of objects can be structured into meaningful sequences or configurations, given certain restrictions, like in derangement problems.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, combination, and arrangement of sets of elements. It’s at the heart of solving problems like the balls and boxes exercise. With combinatorics, we try to find specific arrangements that fit particular rules.
In our case, the task involves determining specific combinations of positioning such that no ball ends up in its matching-colored box. This specific challenge within combinatorics is famously known as the derangement problem.
By using combinatorial principles, you can calculate the total possibilities (via permutations) and subtract or adjust for those that meet certain conditions, offering powerful insight into diverse problems in discrete mathematics.

Discrete Mathematics

Discrete mathematics focuses on the study of structures that are fundamentally distinct or separate. It encompasses a wide range of topics, including graph theory, logic, set theory, and of course, combinatorics. All these give tools and language to model and solve problems.
The derangement problem involving the balls and boxes is a classic example of a problem in discrete mathematics where we look at distinct objects placed into distinct positions without certain overlaps or matches.
This type of problem-solving equips learners with techniques for tackling real-world applications such as networking, computer science, and algorithm design, where understanding unique and specific arrangements is key.