How many ways can 5 boys and 5 girls sit in a row so that no two boys are together?

Answer 1

There are equal numbers of boys and girls, so in order to keep the boys all

separated, there aren't enough girls for any of them to sit together either.

The genders must alternate all the way, end to end. So one gender must

occupy all even-numbered chairs, and the other gender must occupy all

odd-numbered chairs.

-- Number of ways the girls can occupy their 5 chairs = (5 x 4 x 3 x 2) = 120 ways.

For each different way the girls are arranged . . .

-- Number of ways the boys can occupy their 5 chairs = (5 x 4 x 3 x 2) = 120 ways.

-- Total number of ways all ten of them can be arranged = 120 x 120 = 14,400 ways.

-- AND ... each gender can take either the odd or the even seats, and the whole

14,400 can be done either way. If the seats are numbered, then the total number

of ways to assign ten names to the ten numbered seats is 28,800.

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If there were no restriction on who could sit together, then the total number

of ways to arrange ten people would be

10! = (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2) = 3,628,800 ways.