Question

Three boys and three girls are to sit in a row. Find the probability that

i. The boys and girls alternate.

ii. The boys and girls sit together.

iii. Two specific girls sit next to one another.

Please provide full working with correct answer and clear explanation

Answer 1

(i) 3 girls can sit in 3! ways. 3 boys can seat alternatively among the girls in 3! ways. Moreover, we need to consider the fact that either a boy sits in the first seat or a girl sits in the first seat. so there are 2 such cases. So total number of ways = and 6 people can sit in 6! ways.

So probability =

(ii) Let boys be BBB girls be GGG. They sit together. BBBGGG or GGGBBB

So we consider BBB as a single person and GGG as a single person. so 2 persons can sit in 2! ways. Now 3 boys can sit among themselves in 3! ways, similarly for girls. so total number of ways is

so probability is =

(iii) Let two specific girls be A and B. Then AB sit next to one another. So here we consider AB as a single person.b Then remains 6-2=4 persons. So we consider here 4+AB=5 persons. They can sit in ways. and AB can sit together in 2! ways (basically the pattern for A, B is AB or BA) so total number of ways =

Now 6 persons can sit in ways. So probability is =