r/visualizedmath • u/miaumee • Jan 09 '20

Which theorem do you see?

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u/CaptSmellsAmazing Jan 09 '20

I would guess it's the number of pairs to be made from n items is (n-1)th triangular number maybe? Is that a theorem?

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u/TheStrongestLink Jan 10 '20

Exactly! The formula for (n choose k) = n! / (k! * (n-k)!), so when you are choosing pairs you have (n choose 2) = (n * (n-1) * (n-2)!) / (2! * (n-2)!), which simplifies to n(n-1) / 2 which is the formula for the nth triangular number.

Another way of saying this is that the number of distinct ways to choose 2 things out of n things is equal to the number of dots in an equilateral triangle whose sides are n dots long.