Step 1 : Understanding the Question:
This problem deals with permutations, specifically involving constraints. We are asked to arrange 7 people (4 boys and 3 girls) in a row, but with the specific condition that the 3 girls must always stay together in a single block. In combinatorics, this is known as the "string method" or "tie method." Instead of treating everyone as individuals, we group the restricted items together to simplify the arrangement process. This ensures the constraint is met while allowing us to calculate all possible configurations.
Step 2 : Key Formulas and approach:
The number of ways to arrange '$n$' distinct objects is $n!$ (factorial).
Total Arrangements = (Ways to arrange groups) $\times$ (Ways to arrange items within the group).
The approach is to treat the 3 girls as a single unit. We then arrange this unit along with the 4 boys, and finally, arrange the girls among themselves inside their block.
Step 3 : Detailed Explanation:
Step 1: Group the 3 girls together and treat them as 1 single "entity" or "block."
Step 2: Count the total number of units to be arranged. We have 4 boys + 1 girl-block = 5 units.
Step 3: Arrange these 5 units. The number of ways to arrange 5 units is $5!$. $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.
Step 4: Now, consider the arrangements inside the girl-block. The 3 girls can change positions among themselves. The number of ways to arrange 3 girls is $3!$. $3! = 3 \times 2 \times 1 = 6$.
Step 5: Multiply the two results to get the total number of ways. Total ways = $120 \times 6 = 720$.
Step 4 : Final Answer:
The total number of ways to seat them is 720.