Question

You are inviting 4 guests (including yourself) to have dinner at home. Your dining table can seat 6.

How many different seating arrangements could you have?

My Ans:
6! / (6-4)!

1 thing puzzles me is that unlike other questions this time the seats are more than the people.

Answer 1

6!/2!

Answer 2

So is 6!/(6-4)! ?

Answer 3

yes

Answer 4

The thing that puzzle me is more seats than the people avaliable.

Answer 5

Usually it's the other way.

Answer 6

actually, 6!/2! would be the answer if the seats are placed in a row.

in circular permutation, there are (n-1)! ways to permute n people.

Answer 7

hold on...

Answer 8

Omy

Answer 9

forget circular permutation for a moment. In linear permutation, do you see why it's 6!/2!?

Answer 10

Yea, but i also wonder if having less people than the seat will matter or not because the questions i encountered are usually 5 people only 3 seats avaliable. This was different.

Answer 11

it doesn't matter actually. Let's make it simple. Suppose there are 3 seats and only yourself, how many ways can you seat?

Answer 12

3 yes?

Answer 13

Yup

Answer 14

how would you do it mathematically?

Answer 15

Using the formula it will be 3!/(2)!

Answer 16

exactly. treat the two empty seats as same objects. You have 3 objects total. 2 of them are the same, so the total permutation is 3!/2!

Similarly, treat 2 empty seats as same objects. You have 6 objects, two of them are the same, so 6!/2!

Answer 17

now in the case of circular permutation, I was thinking (6-5)! /2! but not so sure.

Answer 18

somehow i feel that should be the answer if I compare with the simplier example above. If there are 3 seats and only yourself, it doesn't matter where you seat, there is only 1 say. That is (3-1)!/2! = 1

Answer 19

so I would say that 6!/2! if this question is an linear permutation application. Otherwise, 5!/2!

Answer 20

why minus 1?