Problem: In how many ways 4 boys and 3 girls stand in a row so that no two girls are together?

Solution:

To solve the problem, we will use the concept of permutations and combinations.

Step 1: Find the total number of ways in which 4 boys and 3 girls can stand in a row without any restriction.

Total number of ways = 7! = 5040

This is because there are 7 people in total and we can arrange them in any order.

Step 2: Find the number of ways in which 4 boys and 3 girls can stand in a row with at least two girls standing together.

There are two cases to consider:

Case 1: Two girls stand together

To determine how many ways two girls can stand together, we can treat them as one unit and find the number of ways in which the 5 units (4 boys, 1 unit of 2 girls) can be arranged in a row. The girls can be arranged in 2! = 2 ways within their unit.

Number of ways = 5! × 2! = 240

Case 2: Three girls stand together

To determine how many ways three girls can stand together, we can treat them as one unit and find the number of ways in which the 4 units (4 boys, 1 unit of 3 girls) can be arranged in a row. The girls can be arranged in 3! = 6 ways within their unit.

Number of ways = 4! × 3! = 144

Step 3: Find the number of ways in which 4 boys and 3 girls can stand in a row with no two girls standing together.

Number of ways = Total number of ways – Number of ways with at least two girls standing together

Number of ways = 5040 – (240 + 144)

Number of ways = 4656

Therefore, there are 4656 ways in which 4 boys and 3 girls can stand in a row so that no two girls are together.