Problem 49 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Suman writes letters to his five friends. The number of ways can be letters be placed in the envelopes so that atleast two of them are in the wrong envelopes are (A) 119 (B) 120 (C) 125 (D) None of these

Step-by-Step Solution

Answer

None of these.

Step 1: Understand the Problem

Suman has five letters and five corresponding envelopes for each letter. We need to find the number of ways to place the letters such that at least two letters are placed into the wrong envelopes.

Step 2: Find the Total Ways without Constraints

First, find the total number of ways to distribute five letters to five envelopes. There are 5 possibilities for the first letter, 4 for the second, 3 for the third, 2 for the fourth, and 1 for the fifth. Hence, the total number of ways is \[5! = 120.\]

Step 3: Calculate Derangement for At Least Two

A derangement means no letter is in its correct envelope. Using the formula for derangements, \[!n = n! \left(\sum_{i=0}^{n} \frac{(-1)^i}{i!}\right),\]we calculate for 5 letters:\[!5 = 5! \left(\frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!}\right) = 44.\]

Step 4: Subtract Derangements from Total

Since derangements ensure all are wrong, and we need at least two wrong, we subtract the derangement from the total possible permutations:\[120 - 44 = 76.\]

Step 5: Final Check with Options

Review the computed answer and compare with given options. The calculated number 76 matches none of the provided options: 119, 120, or 125. Hence, the answer is 'None of these'.

Key Concepts

Derangements

In the world of permutations and combinations, a derangement is a fascinating concept. It deals with the arrangement of objects in such a way that none of them appear in their original position. Imagine you have a set of letters and you want them to be placed in envelopes such that no letter is in its designated envelope. This is a classic example of a derangement.

To find the number of derangements for a set of 'n' items, we use a specific formula:

• The derangement formula: \[!n = n! \left(\sum_{i=0}^{n} \frac{(-1)^i}{i!}\right)\]

This formula takes into account each possible way of ensuring that no item remains in its original slot. It alternates plus and minus terms for each factorial quotient, ensuring the right assessment of each possibility.

For example, with 5 letters, you'd set up your calculation as shown in our original problem, leading to a computation resulting in 44 as the number of possible derangements, meaning 44 ways to rearrange the letters such that none are in their correct envelope.

Factorials

Factorials are a fundamental part of combinatorics. A factorial, denoted by an exclamation point such as \(n!\), is the product of all positive integers less than or equal to \(n\). Factorials are used to calculate permutations, combinations, and derangements. They help determine how many ways you can arrange a set of objects or choose a subset from a larger set.

For instance, the factorial of 5, written as \(5!\), equals:

• \(5 \times 4 \times 3 \times 2 \times 1 = 120\)

This tells us there are 120 different ways to arrange 5 items.

Factorials grow very rapidly with increasing numbers. Just as a fun fact—10! is a staggering 3,628,800! Knowing how to compute them quickly, often by looking up their values or using a calculator, is crucial for solving many permutation and combination problems.

Probability

Probability is the measure of how likely an event is to occur. In the context of permutations and combinations, it often helps us analyze situations where multiple outcomes are possible.

• Probability formula: \[P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\]

In the context of the previous problem with 5 letters, probability could help us measure how likely it is to place all letters incorrectly if desired. However, in our step-by-step solution, we calculated the ways to place them such that at least two were wrong, ensuring we focus on the correct subset of outcomes.

Using probability allows us to estimate the behavior of large systems by considering the likelihood of various arrangements or occurrences. It is a valuable tool for making predictions and informed decisions based on data from permutations and combinations.