Question

In a group of 3 girls and 4 boys, there are two boys \( B_1 \) and \( B_2 \). The number of ways in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but \( B_1 \) and \( B_2 \) are not adjacent to each other, is:

• A. 144
• B. 120
• C. 72
• D. 96

Hint:

For problems involving arrangements with restrictions: - Start by calculating the total number of arrangements without any restrictions. - Then, subtract the cases where the restricted condition is violated (e.g., when \( B_1 \) and \( B_2 \) are adjacent). - Use the principle of inclusion-exclusion if necessary.

Answer 1

The Correct Option is C

We start by treating the girls as a single block since all the girls must stand together. Thus, we have 5 objects to arrange: the girls block and the 4 boys. The total number of ways to arrange these 5 objects is \( 5! \), but we must consider that \( B_1 \) and \( B_2 \) should not be adjacent. First, calculate the total arrangements where all 5 objects are arranged: \[ 5! = 120. \] Next, calculate the number of ways in which \( B_1 \) and \( B_2 \) are adjacent. If they are adjacent, treat them as a single block, so now we have 4 objects to arrange. The total number of ways to arrange these 4 objects is \( 4! \), and within the \( B_1 B_2 \) block, there are \( 2! \) ways to arrange \( B_1 \) and \( B_2 \). Thus, the number of ways in which \( B_1 \) and \( B_2 \) are adjacent is: \[ 4! \times 2! = 24 \times 2 = 48. \] The number of ways in which \( B_1 \) and \( B_2 \) are not adjacent is: \[ 120 - 48 = 72. \] Thus, the answer is \( 72 \).