Problem 37 - A Complete Resource Book in Mathematics for JEE Main

Question

In a group of boys, two boys are brothers and in this group 6 more boys are there. In how many ways they can sit if the brothers are not to sit along with each other (A) 4820 (B) 1410 (C) 2830 (D) None of these

Step-by-Step Solution

Answer

The correct answer is (D) None of these.

Step 1: Calculate Total Seating Arrangements

First, we calculate the total number of ways the 8 boys (2 brothers + 6 more boys) can sit in a row. This can be calculated using the factorial formula for permutations:\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \]

Step 2: Calculate Arrangements with Brothers Together

If the two brothers are considered as a single unit, they can sit together in different ways. By treating them as a single 'block' along with the other 6 boys, we have 7 blocks to arrange. The number of arrangements of these blocks is calculated as:\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 504 \]Within the 'brother block', the two brothers can switch places with each other, giving us:\[ 2! = 2 \]Thus, the number of ways they can sit together is:\[ 7! \times 2! = 504 \times 2 = 1008 \]

Step 3: Subtract Arrangements with Brothers Together from Total Arrangements

Now, subtract the number of seating arrangements where the brothers are together from the total arrangements:\[ 40320 - 1008 = 39312 \]

Step 4: Verify Possible Answers

Compare the computed result (39312) with the given options. Since 39312 is not listed among the answer choices, the correct answer must be (D) None of these.

Key Concepts

Permutations

Permutations refer to the concept of arranging a set of items in a specific order. In simpler terms, it's about finding how many different ways you can arrange things where the order matters. For instance, if you have 3 different colored balls – red, blue, and green – a permutation would consider "red, blue, green" different from "green, blue, red." It's all about sequence. In mathematics, we often use permutations when dealing with problems that involve arranging people, objects, or numbers in various sequences.

• Key point: Order matters in permutations.

In the seating arrangement problem, we calculate how 8 boys can sit in different sequences. This involves using permutation since we're arranging boys in a specific order.

Factorial

Factorial is a fundamental concept in combinatorics, often used to calculate permutations. The notation "n!" represents the factorial of a number, involving multiplying a number by all the integers below it, down to 1.For example, the factorial of 4, written as "4!", is calculated as: \[4! = 4 \times 3 \times 2 \times 1 = 24\]Factorials grow very fast with even small increases in "n." They are incredibly useful for counting possible arrangements of a set of items.

• Key tip: Factorials are always positive integers.
• Remember: 0! is defined as 1.

In the exercise, factorials are used to find the total combinations of seating arrangements. Specifically, the number of ways to seat all 8 boys is represented by "8!."

Seating Arrangements

Seating arrangements are a practical application of permutations in real life, often explored in combinatorics. They involve calculating the number of different ways people or items can be arranged around a table or in a line, adhering to specific rules. In our exercise, we focus on arranging boys in such a way that two brothers do not sit next to each other. It involves quickly calculating how to position the remaining boys around them.

• Premise: The brothers form a 'block' when they sit together.
• We treat the brothers as a single unit to simplify calculations.

First, calculate the total arrangements of all boys. Then determine how the brothers can sit together by treating them as one unit. The problem-solving step involves subtracting the arrangements of this 'blocked' scenario from the total arrangements to prevent them from sitting together. This type of problem is frequent in puzzles and competitions, enhancing analytical skills.