Problem 17 - Algebra for JEE ADVANCED

Question

Total number of words that can be formed using all letters of the word 'BRIJESH' that neither begins with 'I' nor ends with ' \(\mathrm{B}^{\prime}\) is equal to a. 3720 b. 4920 c. 3600 d. 4800

Step-by-Step Solution

Answer

3720

Step 1: Determine the Total Number of Arrangements

The word 'BRIJESH' consists of 7 different letters: B, R, I, J, E, S, H. Therefore, the total number of ways to arrange these letters is 7!, which equals \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\).

Step 2: Calculate Arrangements Starting with 'I'

If the word starts with 'I', then we are left with 6 letters to arrange: B, R, J, E, S, H. The number of such arrangements is 6!, which equals \(6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).

Step 3: Calculate Arrangements Ending with 'B'

If the word ends with 'B', then we are left with 6 letters to arrange: R, I, J, E, S, H. The number of such arrangements is 6!, which equals 720 (same calculation as Step 2).

Step 4: Calculate Arrangements Starting with 'I' and Ending with 'B'

If the word starts with 'I' and ends with 'B', then we have 5 letters to arrange in the middle: R, J, E, S, H. The number of such arrangements is 5!, which equals \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

Step 5: Apply Principle of Inclusion-Exclusion

To find the total number of undesirable words (starting with 'I' or ending with 'B'), use the principle of inclusion-exclusion. Compute: \(720 + 720 - 120 = 1320\). This is the count of arrangements that start with 'I' or end with 'B'.

Step 6: Calculate the Desired Number of Arrangements

Subtract the number of unwanted arrangements from the total arrangements: \(5040 - 1320 = 3720\). This gives the number of arrangements that neither start with 'I' nor end with 'B'.

Key Concepts

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a fundamental concept in combinatorics used to count the number of elements in the union of overlapping sets. It helps us avoid double-counting when the overlap is a concern. For example, if we want to calculate how many ways we can arrange the letters of a word under certain conditions, we might need to consider cases that violate each condition separately and together.

In the context of the given exercise, we have two constraints:

• The word must not start with 'I'.
• The word must not end with 'B'.

Applying inclusion-exclusion, we calculate the total unwanted arrangements starting with 'I' and those ending with 'B', then subtract those where both conditions occur simultaneously. This ensures that each incorrect arrangement is only counted once. The formula is:\[|A \cup B| = |A| + |B| - |A \cap B|\] where \(|A|\) is the count of words starting with 'I', \(|B|\) is the count of words ending with 'B', and \(|A \cap B|\) is the overlap of starting with 'I' and ending with 'B', which are subtracted out to avoid overcounting.

Factorial

Factorial is a mathematical operation that multiplies a given number by every positive integer less than itself. It's represented by an exclamation mark (!). Factorials play a crucial role in permutations and arrangements, especially when calculating the number of ways to arrange a set of distinct items.

For any number \(n\), the factorial is defined as:\[n! = n \times (n-1) \times (n-2) \times \ldots \times 1\]In the given exercise, the word 'BRIJESH' consists of 7 distinct letters, thus the total number of permutations of these letters is given by \(7!\). Calculating \(7!\) yields the result of 5,040 possible arrangements, assuming no restrictions.

Factorials grow very rapidly with increasing numbers, and they are foundational in calculating arrangements, as seen here. They help us understand all possible configurations for a group of unique elements.

Arrangement Problems

Arrangement problems involve finding the number of different ways to order or structure a set of items. They are a key topic in combinatorics and require a good understanding of permutations. Such problems are often framed in terms of arranging letters, people, or other distinct objects.

In the exercise provided, the task is to arrange the seven distinct letters of the word 'BRIJESH' in such a way that certain conditions are met. The original condition was to count arrangements which do not begin with 'I' and do not end with 'B'. These kinds of constraints are common in arrangement problems as they often aim to find the number of valid configurations amidst constraints.

A typical approach to solve these problems involves:

• Determining the total possible arrangements using factorials.
• Calculating arrangements that violate each constraint.
• Applying principles like inclusion-exclusion to accurately count valid arrangements.

Understanding these steps can simplify solving complex arrangement problems by breaking them down into manageable parts.