Question

What's the relation between 9C3 and the 7th Tetrahedral number. They both evaluate to 84... but is there something that maps from one to another?

I am trying to find a way of mapping each of the combinations from 9C3 to a point on the n=7 tetrahedral number.

Answer 1

A few thoughts...

Explanation:

We can note that the tetrahedral numbers `1, 4, 10, 20, 35, 56, 84,...` occur as the fourth diagonals in Pascal's triangle...

The relation between Pascal's triangle and the binomial coefficients becomes clear when you take `(a+b)^n`, write it out as a product of `n` copies of `(a+b)` then think about the choices `a`/`b` from each binomial and how they relate to choosing a left or right branch as you move down Pascal's triangle from the top.

That the fourth diagonal consists of the tetrahedral numbers is clear, since it is a sequence with differences being triangular numbers - which come from the third diagonal. Those triangular numbers are a sequence with differences from the second diagonal, which consists of each natural number in turn, etc.

So the `n`th tetrahedral number is `""^(n+2)C_3`