10 chairs are numbered and arranged in 2 rows as shown in the following diagram.
In how many ways can 9 people, comprising 7 men and 2 women, be seated
(i) without restriction?
(ii) if the 2 women must not sit in the same row?
(iii) if the 2 women must sit together in the same row?
Later, these 9 people are split into 3 groups of 3 people each. In how many ways can this be done?
(i) 10! = 3628800
Note: that an empty seat is a distinct object, treat the empty seat as another person.
women: 2 vs non-women: 8
Total arrangement: 2C1 x 8C4 x 5! x 5! = 2016000
choosing 1 woman to first row: 2C1
choosing non-women to first row: 8C4
No. of arrangement in 1st row: 5!
No of 2nd row arrangements: 5!
Case I (2 women first row) and Case II (2 women second row)
= 2C2 x 8C3 x 4! x 2! x 5! x 2!
Choosing non-women to the first row: 8C3
Arrangements for first row: 4! x 2!
Arrangements for second row: 5!
2 cases: 2
Splitting them into 3 groups of 3 people
Choosing 9 people into the first group: 9C3
Choosing remaining 6 people into the second group: 6C3
Choosing the remaining 3 people into the 3rd group: 3C3
Three groups of the same size: divide by 3!