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?
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.
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