Question

Suppose that I am seating ten of my friends around a circular table. There are 5 males and 5 females, and all distinguishable. Suppose that I am taking a photo with them, i.e. including myself. And I want to arrange us into a row of 5 in front and a row of 6 at the back. Suppose that I want to stand next to (i.e. to the left or right of) my best friend for this picture. How many such arrangements are there?

Answer 1

The number of ways to make the arrangement is computed here as:

= Number of ways such that me and friend are in front row of 5 + Number of ways such that me and my friend are at the back row of 6

In each of the above 2 term, me and my friend are considered one group as we are together. The number of permutations within them could be 2! = 2 though.

Therefore, the number of ways now are computed here as:

= Number of ways to select 6 people for the back row * Number of permutations of those 6 people at back row * Number of permutations of 4 groups in front * Number of permutations of me and friend + Number of ways to select 5 people for front row * Number of ways to arrange those 5 people * Number of permutations of 5 groups at back row * Number of permutations of me and friend.

These are the required number of ways here.