A person wishes to make up as many different parties as he

 can out of his 20 friends such that each party consists of the

 same number of persons. The number of friends he should  

invite at a time, is  

Suppose he invites r friends at a time. Then the

 total number of parties is 20 Cr. We have to find the maximum 

value of  $20 \mathrm{Cr}\text{,}$ which is for r $r=10$ (if 11 is even, then ${ }^{\prime \prime }{C}_r$, is maximum for r = n/ 2).

 Hence, he should invite 10 friends at a  ime in order to form the

 maximum number of parties .