Three boys and four girls line up randomly in a row What is

Three boys and four girls line up randomly in a row. What is the probability that

a.) All the boys are on one side?

b.) The boys and girls are in alternate positions.

a) No.of girls = 4 no.of boys = 3

Total no.of members = 7

7 members can be arranged in 7! ways = 5040.

All the boys are on one side can happen in two ways at the starting or at the end

If all the boys are in the starting then arrangement is = BBBGGGG .

These boys can be arranged in 3! ways and 4 girls can be arranged in 4! ways = 3! * 4! = 144

If all the boys are at the end then arrangement is = GGGGBBB .

These boys can be arranged in 3! ways and 4 girls can be arranged in 4! ways = 3! * 4! = 144

Therfore total no.of ways that boys can be on one side = 144+144 = 288

Therefore P( All the boys are on one side ) = 288/ 5040 = 0.057

b) The boys and girls can be arranged in alternative possition in only one possible way GBGBGBG

3 boys can be arranged at 3 places in 3! ways and 4 girls can be arranged in 4 places in 4! ways.

Therefore P(The boys and girls are in alternate positions. ) = (3! *4!) / 5040 = 144/5040 = 0.02857