Definitions, by order of appearance

[n]d[k]:

a roll of n k-sided dice, where you add the pips together; when n is 1 it is often omitted

n-simplex:

the simplest shape in n dimensions that bounds some “volume”

bijection:

a 1 to 1 matching of items in one set to items in another. It is often used to prove that the sets are of the same size; often because sizes of certain sets are easy to directly compute