Problem 13 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

There are 4 envelopes with addresses and 4 concerning letters. The probability that letter does not go into concerning proper envelope is (a) \(19 / 24\) (b) \(21 / 23\) (c) \(23 / 24\) (d) \(1 / 24\)

Step-by-Step Solution

Answer

The correct answer is actually \(3/8\) and is not listed among the options.

Step 1: Understand the Problem

We have 4 distinct envelopes, each with a unique address, and 4 letters that are intended for each of these envelopes. We are asked to determine the probability that none of the letters ends up in the correct envelope.

Step 2: Calculate Total Possible Arrangements

Calculate the total number of possible ways to arrange 4 letters into 4 envelopes. Since this is simply an arrangement (permutation) problem, we use the factorial of the number of items: \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \] So, there are 24 possible ways to place the letters into the envelopes.

Step 3: Determine Derangements

A derangement is a permutation where none of the objects appear in their original position. For 4 objects, the number of derangements \( !4 \) is calculated using the formula: \[ !n = n! \sum_{i = 0}^{n} \frac{(-1)^i}{i!} \] For 4 objects, we have: \[ !4 = 24 \left( \frac{(-1)^0}{0!} + \frac{(-1)^1}{1!} + \frac{(-1)^2}{2!} + \frac{(-1)^3}{3!} + \frac{(-1)^4}{4!} \right) \] \[ = 24 \left( 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} \right) \] \[ = 24 \times \frac{9}{24} = 9 \] So there are 9 derangements of the 4 letters.

Step 4: Calculate Probability

The probability that none of the letters ends up in the correct envelope is the number of derangements divided by the total number of arrangements: \[ \frac{!4}{4!} = \frac{9}{24} = \frac{3}{8} \] This means that none of the options provided in the question are correct.

Key Concepts

Derangements

Derangements are a fascinating concept in probability and combinatorics. They occur when no item appears in its original position. For example, suppose there are four letters and envelopes, each associated with a specific letter. A derangement would mean that none of the letters are in their intended envelopes. In mathematical terms, if we have "n" items, a derangement is a permutation where no item is in its starting position.

We use the following formula to calculate the number of derangements for "n" items:

• \[ !n = n! \sum_{i=0}^{n} \frac{(-1)^i}{i!} \]

This formula considers the total permutations and subtracts the arrangements that include at least one item in its original place.

For four items, the number of derangements can be calculated as follows:

• \[ !4 = 24 \left( 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} \right) = 9 \]

This means there are 9 ways to arrange four letters such that none of them ends up in the correct envelope. This concept is particularly useful in probability exercises involving misplaced items.

Permutations

Permutations are arrangements of items in a particular order. In probability and combinatorics, permutations are used to find how many ways items can be ordered. If you have "n" objects, the number of permutations is given by \( n! \), where "!" denotes the factorial operation.

Factorials are calculated by multiplying the number down to 1. For instance, \( 4! \) means \( 4 \times 3 \times 2 \times 1 = 24 \). This computation tells us there are 24 different ways to arrange four distinct items. Permutations are crucial in scenarios where the order of arrangement matters.

In the activity involving envelopes and letters, we applied permutations to understand the different ways the letters can be inserted into the envelopes. We calculated a total of 24 permutations. This provided the groundwork for further investigating how many of those permutations were derangements, where letters did not end up in their intended envelopes.

Combinatorics

Combinatorics is a branch of mathematics focusing on the study of counting, combinations, and permutations. It helps us solve problems related to arranging and choosing items in sets. Whether we are interested in arranging a group of letters or choosing a team, combinatorics provides the necessary tools.

A key part of combinatorics involves understanding how to efficiently count possible outcomes in a situation. This involves concepts like permutations, as discussed earlier, and combinations, which account for selecting items without regard to order. In our envelope and letter problem, combinatorics allowed us to calculate permutations and derangements to find a no-right-envelope scenario.

Combinatorics connects directly to probability, as it helps establish all possible outcomes and their likeliness in an event. This systematic counting framework underpins the calculation of probability, offering insight into questions like the probability that none of the letters ends up in the correct envelope, as explored with our problem.