I you are in a group of 'x' people - there are x-1 people you can choose your pair from.

So you can choose possible pairs.

For 4 people we have - possible pairs.

Because there are x people in the group you can do it x times, so we have: possible pairs:

For 4 people we have - 3 x 4 = 12 pairs. (the denominator)

Now if you where not there you would not be chosen - so there are now x-1 people playing - substitute x-1 in to the previous formula to get the number of pairs not including 1 person.

We get:

Numerator:

Denominator:

If you can simplify this factorial stuff pls do.

This gives probabilites of not being chosen as:

4 people: 3/12

5 people: 12/30

6 people: 30/60

7 people: 60/105

Now to generalise:

n people - choosing m (eg 5 people chosing 2 people)

We get:

Numerator:

Denominator:

i thonk there is an error here but im tired - i look again soon.