Hello, Aaron!

You have 20 people sitting around a table.

You want to form a committee of 3 people.

What is the probability/chance that at least 2 of the people are sitting together?

Suppose the people are: .

. . seated around the table in that order.

There are: . possible committees.

We must consider two cases:

. . (1) The 3 people are sitting together.

. . (2) Exactly 2 of the 3 people are sitting together.

(1) 3 people are together.

There are 20 ways: .

(2) Exactly 2 people are together.

There are are 20 choices for the Pair: .

There are 18 choices for the third person.

But he/she must not be adjacent to the Pair.

Hence, there are only 16 choices for the third person.

There are: . ways.

Therefore, there are: . ways.

. .