# Find the number of ways in which 5 boys and 5 girls may be seated in a row so that no two girls and no two boys are together.

Permutations and Combinations are very important concepts in mathematics that deals with arrangements and selections of various kinds of objects. This field of mathematics has many applications in other fields as well, which include science and engineering, and also has daily life applications. Let's solve a problem to make the concept more clear.

## Answer: The number of ways in which 5 boys and 5 girls may be seated in a row so that no two girls and no two boys are together is 10! - 5! × 6!.

Let's understand the solution.

**Explanation:**

According to the given information, we have 5 girls and 5 boys.

The number of boys can be seated in a row in ^{5}p_{5} = 5! ways

The girls can fill the gaps in ^{6}P_{5} = 6! ways.

The number of ways in which no two girls sit together = 5! × 6!.

the total number of permutations to arrange 5 girls and 5 boys = ^{10}P_{10} = 10! ways.

Total number of ways in which no girls are together = Total number of arrangements – Number of arrangements in which all the girls are together