In how many different ways can 4 girls and 5 boys be arranged in a single row so that all 4 girls always stand together as one consecutive group?

Introduction / Context:
This question is a classic arrangement problem in permutations where a subset of people must always stand together. The 4 girls must be treated as a single block, and within that block they can be permuted among themselves. This checks understanding of grouping in permutations.

Given Data / Assumptions:

• There are 4 distinct girls and 5 distinct boys.
• We arrange all 9 people in a single row.
• All 4 girls must stand together as one consecutive group.
• Different orders of the same individuals count as different arrangements.

Concept / Approach:
We first treat the 4 girls as a single super person or block. Then we count permutations of this block together with the 5 boys. After that, we account for the internal permutations of the 4 girls inside the block. The total arrangements are the product of these two counts.

Step-by-Step Solution:
Step 1: Consider the 4 girls as a single block G.Step 2: We now have 5 boys plus the one block G, making 6 entities in total.Step 3: The number of ways to arrange these 6 entities in a row is 6! (because all are distinct at this level).Step 4: Compute 6! = 720.Step 5: Inside the girls block, the 4 distinct girls can be arranged among themselves in 4! ways.Step 6: Compute 4! = 24.Step 7: Combine the counts: total arrangements = 6! * 4! = 720 * 24 = 17280.

Verification / Alternative check:

Why Other Options Are Wrong:

• 18000, 17829 and 18270: These values do not match the exact product of 6! and 4!, and likely result from arithmetic mistakes or miscounting entities such as using 7! or only 5! instead of 6!.

Common Pitfalls:
Many students forget to multiply by the internal permutations of the 4 girls and stop at 6!, which undercounts. Others mistakenly treat girls and boys as identical groups or miscount the total number of entities to arrange as 9 instead of 6 after grouping. It is important to clearly separate the outer arrangement of blocks and individuals from the internal arrangement within the block.

Final Answer:
The number of ways to arrange the 4 girls and 5 boys with all girls together is 17280.