Question

6 men and 8 women are going to be assigned to a specific row of seats in a theater. If the 14 tickets for the numbered seats are given out at random, find the probability that exactly one womb is in one of the first 5 seats.

Answer 1

If exactly one woman is to sit in one of the first 5 seats, then it means that 4 men completes the first 5 seats.
No of ways 4 men can be selected from 6 men = 6C4 = 15
No of ways 4 men can sit on 5 seats = 5P4 = 120
No of ways 1 woman can be selected fom 8 women = 8C1 = 8
No of ways 1 woman can sit on 5 seats = 5P1 = 5

No of ways that exactly one woman is in one of the first 5 seats = 15 * 120 * 8 * 5 = 72,000

No of ways 14 people can be arranged in 14 seats = 14!

Probability that exactly one woman is in one of the first 5 seats = 72,000 / 14! = 0.0000008259 = 0.000083%