9 people sit around 1 rectangular table and 1 circular table.each table has 6 identical seats,other table is occupied only if the first table is fully occupied.how many arrangements are possible?
Answers
This is a question you will get under your permutation and combination lesson.
We can divide this question into 2 parts.
Part 1 : 6 people can sit around the rectangular chair and the other three can sit around the circular chair.
Part 2 : 6 people can sit around the circular chair and the other three can sit around the rectangular chair.
We can get the answer by adding the answers of part 1 and 2.
Part 1 :
• No of ways 6 people can seat in 6 seats in a rectangular chair is 6! (6P6)
• No of ways 3 people can seat in 6 seats in a circular chair is
6C3 x (3-1)!
It is important to remember two things here,
1. You have to be mindful that before arranging seats, people have to choose 3 seats out of 6 in the circular chair (That is combination).
When arranging those three people, we have to take the possible number of ways ordering these 3 (Permutation). Since this is a circular order, we have to take the Permutation as (3-1)!
Part 1 = 6! x 6C3 x (3-1)! -------------------------(1)
Part 2 :
• No of ways 6 people can seat in 6 seats in a circular chair is (6-1)!
• No of ways 3 people can seat in 6 seats in a rectangular chair is
6C3 x (3)!
Part 1 = (6-1)! + 6C3 x (3)! -------------------------(2)
Answer is = (1) + (2)
= (720 x 20 x 2) + (120 x 20 x 6)
= (720 x 20 x 2) + (720 x 20)
= (720 x 20) (2+1)
= 720 x 20 x 3
= 43200