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One day i got bored and was looking at my arms birthmarks and decided to count the amount of unique triangles i could make with just 4 birthmarks, lets call them dots in the puzzle.

To clarify: a unique triangle uses three dots and at least 1 of them is different from the other triangles. (4 dots can make 4 triangles; 1234: 123,124,134,234)

After i was done i added a fifth dot and started counting again..

I figured there should be a formula to calculate it, and so it got me started on creating that formula. I had some fun creating it and i hope you will as well.

Create the formula and tell how many unique triangles 13 dots can make

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  • $\begingroup$ for me, creating a formula is like a logic puzzle including math. $\endgroup$
    – sjaak bakker
    Jan 27, 2016 at 7:48

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The answer is:

286

Why?

Using the general formula $C(n,r)=\frac{n!}{r!\cdot (n-r)!}$, you plug in $13$ for $n$ and $3$ for $r$ to get 286. $n$ is the set size and $r$ is the subset size.

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  • $\begingroup$ Not really created yourself ;-), but the formula is obviously correct and the answer accepted. $\endgroup$
    – sjaak bakker
    Jan 26, 2016 at 13:37
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Assuming no three dots are collinear, the formula for $n \ge 3$ dots is

$\binom{n}{3} = \frac{1}{6}(n-2)(n-1)n$

So for 13 dots we get

$\frac{1}{6} \cdot 11 \cdot 12 \cdot 13 = 286 \mbox{ triangles}$

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  • $\begingroup$ Great job Will, took me longer to create this same formula. $\endgroup$
    – sjaak bakker
    Jan 26, 2016 at 13:41
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    $\begingroup$ By some definitions, it doesn't even matter if they are collinear. You just get a degenerate triangle with angles 0, 0, 180. $\endgroup$
    – Darrel Hoffman
    Jan 26, 2016 at 14:59

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