In how many ways can 5 boys and 3 girls be seated in a row such that no two girls are together?





We have a total of 8 people who need to be seated in a row. The number of ways to arrange them is:

8! = 40,320



If two girls sit together, we can consider them as one entity. Then we have 7 entities to arrange in a row, which can be done in:

7! × 2! = 10,080

The factor of 2! accounts for the number of ways to arrange the two girls within their entity.



If three girls sit together, we can also consider them as one entity. Then we have 6 entities to arrange in a row, which can be done in:

6! × 3! = 2,160

The factor of 3! accounts for the number of ways to arrange the three girls within their entity.



The number of ways with two or three girls together is:

10,080 + 2,160 = 12,240

So the number of ways with no two girls together is:

40,320 - 12,240 = 28,080

Therefore, there are 28,080 ways to seat 5 boys and 3 girls in a row such that no two girls are together.