Question

The number of ways 4 boys and 3 girls are to be arranged in a row so that all 3 girls are not together, is equal to:

• A. 2320
• B. 4320
• C. 4920
• D. 1440

Hint:

Note the wording: "all 3 girls are not together" means you only subtract the case where the block of 3 is intact. It still allows for 2 girls to be together. If the question said "no two girls are together," you would use the Gap Method instead.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To find the number of ways where all 3 girls are not together, it is easiest to subtract the cases where they are together from the total possible arrangements.
Step 2: Key Formula or Approach:
Total arrangements of \( n \) distinct objects = \( n! \). Arrangements with items together = (Grouped item treated as 1 unit) \( \times \) (Internal arrangements of that unit).
Step 3: Detailed Explanation:
Total number of children = \( 4 \text{ boys} + 3 \text{ girls} = 7 \). Total arrangements = \( 7! = 5040 \). Now, find arrangements where all 3 girls are together: Treat the 3 girls as a single "block". Now we have 4 boys + 1 block = 5 units. Ways to arrange 5 units = \( 5! = 120 \). Ways to arrange the 3 girls inside the block = \( 3! = 6 \). Total "together" cases = \( 120 \times 6 = 720 \). Number of ways where all 3 girls are not together: \[ 5040 - 720 = 4320 \]
Step 4: Final Answer:
The number of ways is 4320.