Combinatorics in a restaurant

Question

In a restaurant menu there are 6 types of drinks : Coca cola , lemonade , sprite , wine , tea and diet sprite .

How many people need to order a drink to ensure that at least one drink would be ordered by two people ?

I was thinking about 6 above 2 which is : 6!2! / 4! = 15 . But it doesn't makes sense .

Can someone please verify that ?

Answer 1

A direct application of the weak form of the pigeon-hole principle yields the result immediately.

Is it possible for six people to all be drinking different things?

Yes. Adam could drink Cocacola, Billy could drink lemonade, Charles could drink sprite, David could drink wine, Emmanuel could drink tea, and Frank could drink diet sprite.

If we add a seventh person, is it still possible for them all to be drinking different things?

No. Once we have a seventh person, there must be someone drinking the same thing as someone else. Why? Well... suppose that all seven of them were in fact drinking different things. But then we would have seven different drinks, but there are only six types of drinks available, a contradiction.