In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, 3 in the front and 4 at the back? How many seating arrangements are possible if 3 girls sit together in a back row on adjacent seats?Now, if all the seating arrangements are equally likely, what is the probability of 3 girls sitting together in a back row on adjacent seats?

Explanation

We have 14 seats in two vans and there are 9 boys and 3 girls. The number of ways of arranging 12 people on 14 seats without restriction is
`.^(14)P_(12)=(14!)/(2!) = 7(13!)`
Now, the number of ways of choosing a van is 2. The number of ways of arranging girls on three adjacent seats in selected van is `2(3!)`. The number of ways of arranging 9 boys on the remaining 11 seats is `.^(11)P_(9)`. Therefore, the required number of ways is
`2xx(2xx3!)xx .^(11)P_(9)= (4!11!)/(2!) = 12!`
Hence, the probability of the required events is `(12!)/(7 xx 13!) = (1)/(91)`