There are 14 chairs and 8 people to be seated. But among the 8. three will be seated together:
So 5 people and (3) could be considered as 6 entities:
Since the order matters, we have to use permutation:
¹⁴P₆ = (14!)/(14-6)! = 2,162,160, But the family composed of 3 people can permute among them in 3! ways or 6 ways. So the total number of permutation will be ¹⁴P₆ x 3!
2,162,160 x 6 = 12,972,960 ways.
Another way to solve this problem is as follow:
5 + (3) people are considered (for the time being) as 6 entities:
The 1st has a choice among 14 ways The 2nd has a choice among 13 ways The 3rd has a choice among 12 ways The 4th has a choice among 11 ways The 5th has a choice among 10 ways The 6th has a choice among 9ways
So far there are 14x13x12x11x10x9 = 2,162,160 ways But the 3 (that formed one group) could seat among themselves in 3! or 6 ways:
Total number of permutation = 2,162,160 x 6 = 12,972,960
