1. You invite n guests to a party, none of whom know each other when the party s

Question

1. You invite n guests to a party, none of whom know each other when the party starts.  
a) If you want to introduce your guests to each other two at a time, how many introductions do you need to make in all? Find a formula in terms of n.
b) Is it possible to introduce some of the guests to each other so that in the end, no two guests know the same number of people at the party? Why or why not?
c) Is it possible to introduce some of the guests to each other so that in the end, exactly two guests know the same number of people at the party? Why or why not?

Explanation / Answer

Answer

Given Condition is Invite guests to a party, none of whom know each other when the party starts.

Formula for the term n

Let there are 5 guests in the party so we need 4+3+2+1 introduction. Such that,

Guest 1 meets other 4 (2,3,4 and 5)

Guest 2 meets other 3 (3,4,5) but not guest 1. Since, they are already introduced.

Guest 3 meets other 2 (4,5) but not guest 1,2. Since, they are already introduced.

Eventually, guest 5 got introduced to all.

Therefore, number of introductions is 4+3+2+1=10

Thus, it can be generalized as [ n(n-1)/2 ] introductions. Where n is the number of guests.

(b) No, it is not possible. Since, the introduction takes place in pairs. So, for any combination of introduction, there always will be at least one pair who is being introduced to same number of guests.

( c ) Yes, it is possible. It can be achieved by isolating two guests from introduction to each other but introduce them to everyone separately. Later, introduce all remaining guests(except two) to each other.

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