# Mike is going to sit at a round table with 5 of his friends. How many possible pairs of friends could Mike be sitting between?

Mike is going to sit at a round table with 5 of his friends. How many possible pairs of friends could Mike be sitting between?

There are 6 people in total and so there are 5! possible seating arrangements. How can I place Mike between pairs. How can I count it. Any hints about this would be greatly appreciated.

Is it 10? Since there are 5 slots for Mike to sit and we are talking about pairs, so 5 choose 2?

• Doesn't it ask how many different pairs of people Mike can sit between. So if there are 5 other people call them {a,b,c,d,e}, then Mike can sit between ab, ac, ad, ae, bc, bd, be, cd, ce, de. So wouldn't there be 10 different pairs of people he could sit between? – ddswsd Oct 24 '17 at 0:22
• Am I missing something or doing this completely wrong? – ddswsd Oct 24 '17 at 0:22

Assuming that it doesn't matter who sits to his left and to his right, you are correct: it is simply the number of possible pairs of friends. The round table is just a distraction.

So mike is always going to be between two people. How many different ways of choosing two people out of 5 are there?

Since there are 5 slots for Mike to sit and we are talking about pairs, so 5 choose 2?

This logic is incorrect.

Hint: Try to see how many combinations there are while choosing 2 people out of 5

If P1-Mike-P2 and P2-Mike-P1 count as different combinations,
No. of possibilities for the person sitting on his left is 5 (6 people minus Mike)
and
No. of possibilities for the person sitting on his right is 4 (6 people minus Mike and Person on left).
On multiplying you get: 5*4=20 possible combinations.

If P1-Mike-P2 and P2-Mike-P1 don't count as different combinations,
There would be 20/2=10 combinations possible.

• By convention, in a circular arrangement, only the relative order matters. Seat Mike first. Relative to Mike, there are $5!$ ways to seat the others as we proceed clockwise around the table. Also, while arrangements are affected by who is sitting to Mike's left, which pair of friends sits beside him is not. – N. F. Taussig Sep 21 '18 at 22:24
• Yeah... thanks, I made the necessary edits. Also, could you rephrase the second part of the comment? I can't understand what you mean by "which pair of friends sits beside him is not." – Stephen Allen Sep 22 '18 at 17:10
• What I was trying to convey is that it does not matter if Gary sits to Mike's left and Paul sits to Mike's right or if Paul sits to Mike's left and Gary sits to Mike's right, Mike is still sitting between Gary and Paul even though the arrangement is different. – N. F. Taussig Sep 22 '18 at 18:56
• Okay, thanks... So it is the second case. – Stephen Allen Sep 23 '18 at 16:18