# Dinner party clinks

A few weeks ago I asked:

*“ At a dinner party for 10, each person ‘clinks’ their glass once with each other person. How many clinks?”*

I’ll just leave some space here for you to think about it…

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Got any ideas? Maybe you know the problem already. There is more than one way to think about this one (as there often is). Here’s two…

**1. Purposely double count**

Well, each of the 10 people ‘clink’ with the other 9, so that’s 10 x 9 = 90 clinks, right?

Not quite – we’ve counted each clink twice.

For example, we counted when person 1 clinked with person 2, then we counted again when person 2 clinked with person 1.

So, there were 10 x 9 / 2 = 45 clinks.

Similarly, for 12 people, there would be 12 x 11 / 2 = 66 clinks.

More generally, for n people, there would be n(n-1)/2 clinks.

**2. Count as we go**

Person 1 clinks with 9 others.

Person 2 clinks with 8 people, not counting the already counted clink with person 1.

Person 3 clinks with 7 people, not counting the clinks with persons 1 and 2.

Person 4 clinks with 6 people, not counting the clinks with persons 1 to 3.

Person 5 clinks with 5 people, not counting the clinks with persons 1 to 4.

Person 6 clinks with 4 people, not counting the clinks with persons 1 to 5.

Person 7 clinks with 3 people, not counting the clinks with persons 1 to 6.

Person 8 clinks with 2 people, not counting the clinks with persons 1 to 7.

Person 9 clinks with 1 people, not counting the clinks with persons 1 to 8.

Person 10 has had all their clinks counted already!

So the number of clinks is 9+8+7+6+5+4+3+2+1 = 45

**7 year old Gauss’s formula**

See my earlier post here for the relationship between the above two methods.