Number of Ways to Form Queue with Restrictions

To find the number of ways to form a queue of 4 boys and 3 girls such that all the girls are not together, we can subtract the number of ways in which all the girls are together from the total number of unrestricted arrangements.

Step 1: Calculate the total number of unrestricted arrangements The total number of people is $4 \text{ boys} + 3 \text{ girls} = 7 \text{ people}$. The total number of ways to arrange these 7 people in a queue is given by: $N_{\text{total}} = 7! = 5040$

Step 2: Calculate the number of ways where all 3 girls are together To ensure all 3 girls are together, we can treat the 3 girls as a single unit or block.

• Now, we have 4 boys and 1 block of girls, which makes a total of $4 + 1 = 5$ entities to arrange.
• The 5 entities can be arranged among themselves in $5!$ ways.
• The 3 girls within the block can be arranged among themselves in $3!$ ways.

Thus, the number of ways in which all 3 girls are together is: $N_{\text{together}} = 5! \times 3! = 120 \times 6 = 720$

Step 3: Calculate the required number of ways The number of ways to form the queue such that all the girls are not together is: $N_{\text{not together}} = N_{\text{total}} - N_{\text{together}}$ $N_{\text{not together}} = 5040 - 720 = 4320$

Thus, the correct option is D.