So I was trying to figure out a combinatorics problem involving the number of unique paths between two coordinates (can't move backwards such as from (1,1) to (0,1)) and I got stuck. I decided to draw all the paths between different coordinates to see if I could find a pattern. I started with going from (0,0) to (3,0) and found the number of unique paths. Then I found the number of unique paths going from (0,0) to (3,1) and so on (I did this until (3,4)). Putting these values in a sequence, I got 1, 4, 10, 20, and 35 which are the first 5 tetrahedral numbers. Is there an explanation for this or is it just a coincidence?
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The numbers you're seeing are binomial coefficients. To go from $(0,0)$ to $(3,n)$ you have to take $3+n$ steps: $3$ right and $n$ up. You have to choose which $3$ of the $3+n$ steps that are going to go to the right, and the number of ways to choose is $\binom{3+n}{3}$ (which is $T_{n+1}$ in Wikipedia's notation for tetrahedral numbers).
Hans Lundmark
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