Problem:
In how many ways can 5 boys and 3 girls be arranged so that no two girls may sit together?

Solution:

To solve this problem, we can use the concept of permutations. We need to find the number of ways the boys and girls can be arranged in a line such that no two girls are sitting together.

Approach:

We can break down the problem into smaller steps to find the total number of arrangements.

Step 1: Calculate the total number of ways to arrange the boys and girls without any restrictions.

Since we have 5 boys and 3 girls, the total number of ways to arrange them is given by the permutation formula:

P(8, 8) = 8!

Here, 8! represents the factorial of 8, which is the product of all positive integers from 1 to 8.

Step 2: Calculate the number of arrangements where the girls are sitting together.

Since there are 3 girls, we can consider them as a single entity. So, we have 6 entities in total (5 boys + 1 group of girls).

The number of arrangements where the girls are sitting together is given by the permutation formula:

P(6, 6) = 6!

Step 3: Subtract the number of arrangements where the girls are sitting together from the total number of arrangements.

The number of arrangements where no two girls are sitting together can be calculated by subtracting the number of arrangements where the girls are sitting together from the total number of arrangements:

Total number of arrangements - Number of arrangements where girls are sitting together

= P(8, 8) - P(6, 6)

Final Answer:
By substituting the values in the above equation, we can find the number of ways the boys and girls can be arranged such that no two girls are sitting together.

Visually appealing breakdown of the solution:
- Problem: In how many ways can 5 boys and 3 girls be arranged so that no two girls may sit together?
- Solution approach:
- Step 1: Calculate the total number of ways to arrange the boys and girls without any restrictions.
- Step 2: Calculate the number of arrangements where the girls are sitting together.
- Step 3: Subtract the number of arrangements where the girls are sitting together from the total number of arrangements.
- Final Answer: Evaluate the expression P(8, 8) - P(6, 6) to find the number of ways the boys and girls can be arranged without any two girls sitting together.

⁶C3×3!×5!