The toast formula

Imagine, you’re on a party, everyone has a glass of vine in its hand and everyone raises its glass to everyone other in the room. So if it’s a small party with just 3 people (lets call them A, B and C), A will toast with B and C and B will toast with C – in total, you hear the glasses clinking three times. What, if there`re 23 people, how often will there be a toast?

There are several ways to get the answer. The “brute force” method is counting up

1 person = 0 times
2 people = 1 time
3 people = 3 times
4 people = 6 times
5 people = 10 times
6 people= 15 times
… and so on…

It works, but it costs much time, and is not efficient. So the question is: Can we find a formula to get the answer? We can. Do we want to find it? It’s up to you. But let’s start at the very beginning. We see in the enumeration above, that each person corresponds to a number of clinking glasses. 1 leads to 0, 2 leads to 1, 3 leads to 3 and so on. Let’s think of 5 books in different colors and 5 places in a shelf where we can put them. How many possibilities do we have to order these books? For the first book, we can choose all 5 places in the shelf since we haven’t put a book in yet. For the second book, we just have 4 free spaces – since the first book is already in the shelf. For the third book we have 3 free spaces and so on. 5 possibilities for the first book and 4 possibilities for the second book are in total 5*4 = 20 possibilities for the first two books. For all books, the number of possibilities are 5*4*3*2*1 = 120. Mathematics have another way to write sequences like 5*4*3*2*1, they just write 5! – this is called the factorial of five.
Think of another example: There’s again a shelf with 5 free spots but at this time, the books look exactly the same. As another restriction, we just have two books. How many possibilities of dispositions do we have now? Again, we can put the first book in all 5 places. And we can put the second book in the other 4 places. The difference here is that we don’t distinguish between the books, so it’s the same of the first book is in the first place, the second in the second place or if the first book is in the second place and the second in the first place. There’s also a formula to express this:


The variable n is the number of objects or – in our example, books. N is the number of free places in the shelf. If we are looking for the number of ways, we can arrange 2 books in a shelf with 5 places, we just put in the numbers:


So, there are 10 different ways to place 2 books in 5 places. And this is exactly what we’re looking for to get our toast formula: Here, n is the number of persons we have at the party, k is the number of persons which speak a toast to each other (2).

So, how often will there be a toast if 22 people are in the room?


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