I suspect that
this page from the same site might be a little more helpful. The first line says:
The number of ways of picking k unordered outcomes from n possibilities. Also known as the binomial coefficient or choice number and read "n choose k,"
So, when someone says "52 choose 2" they mean "the number of ways you can choose 2 items from a collection of 52 items when the order you choose them in doesn't matter" (for example, getting dealt a 2 card hand from a deck of 52 cards). The formula there is then what people are using when solving the equations.