Three girls and four boys are seated randomly on a straight bench. What is the probability that the boys sit together and the girls sit together.
To calculate the desired probability we need to calculate the number n of arrangements of the kids such that the girls sit together and the bous sit together and then calculate the total number m of arrangements. Then the probability is n/m.
Since boys can sit to the left or to the right from the girls we have 2 possibilities of arranging full groups. 4 boys in a group can be seated in 4!=24 different ways. The group of 3 girls can be seated in 3!=6 different ways. Hence the number of arrangements in which the two groups are separated is 2246=288.
On the other hand there are 7!=5040 different arrangements of a group of 7 people. Therefore the probability of the two groups being separated is 288/5040=2/35.
JW
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