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.