Determine the total number of proper subsets of the set of letters from the english alphabet a, b, c, . the number of proper subsets is

has ten elements, so any proper subset has at most nine of these elements.

The number of ways of taking any letters from this set is given by the binomial coefficient,



and in particular, the total number of ways to picking proper subsets is



Without computing each term directly, let's instead use a direct result from the binomial theorem, which says



If we replace , then we're left with



We can use this to evaluate our sum directly:

You need to include the original coordinates of the triangle to answer this. and how much does it translate (move left right up down)