A person tries to form as many different parties as he can, out of his 20 friends. Each party should consist of the same number. How many friends should be invited at a time? In how many of these parties would the same friends be found?

Explanation

Let the person invite r number of friends at a time. Then, the number of parties are `.^(20)C_(r)`, which is maximum, when r=10
If a particular friend will be found in p parties, then p is the number of combinations out of 20 in which this particular friend must be included. therefore, we have to select 9 more from 19 remaining friends.
hence, `p=.^(19)C_(9)`.