Question

There are 14 places in the circle. How many ways can 10 boys and 4 girls sit provided that they do not sit together?

Answer 1

To determine the number of ways 10 boys and 4 girls can sit in 14 places in a circle such that the 4 girls do not sit together, we'll follow these steps:
1. Fix the Circle Arrangement for Boys: Since the arrangement is circular, fixing one boy in place will eliminate identical rotations. Thus, we arrange the remaining 9 boys in $9!$ ways.
2. Place the Girls: To ensure the girls do not sit together, we consider the gaps created by the 10 boys. In a circle of 10 boys, there are 10 gaps where the girls can be placed.
3. Select 4 Gaps for the Girls: We choose 4 out of these 10 gaps for the girls. This can be done in $\binom{10}{4}$ ways.
4. Arrange the Girls in the Selected Gaps: The 4 girls can be arranged in the chosen 4 gaps in $4!$ ways.Thus, the total number of ways to arrange the boys and girls under the given conditions is:
$9! \times \binom{10}{4} \times 4!$
Now, let's compute the values:
- $9! = 362,880$
- $\binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210$
- $4! = 24$
Multiplying these values together gives us the total number of arrangements:

$9! \times \binom{10}{4} \times 4! = 362,880 \times 210 \times 24$

Let's compute this step-by-step:

$210 \times 24 = 5040$
$362,880 \times 5040 = 1,830,758,400$

Therefore, the number of ways 10 boys and 4 girls can sit in 14 places in a circle such that the girls do not sit together is $1,830,758,400$.