Question

In how many ways can 10 people be seated in a row so that a certain pair of

them are not next to each other?

Three persons enter into car, where there are 5 seats. In how many ways can

they take up their seats?

Answer 1

1)To begin we calculate number of ways we can seat this pair. If on person of pair seated on edge of row, 8 seats available for another person in pair, otherwise there are 7 seats for second person. So we have (2*8+8*7) ways to seat this pair. And for each way of placing pair we should distribute 8 seats between 8 people without any conditions which gives us 8! ways of placing. So there are (2*8+8*7)8!=2,903,040 ways.

2)If car means automobile and there are must be driver, then number of ways is 3*4*3=36 (3 ways to choose driver and 4!/(4-2)!=4*3 is number of ways to distribute 4 seats between 2 people. If car means railroad car and there are not any conditions of seating, number of ways will be 5*4*3=60.