Probability and optimization
I am given the entropy function of S(P(x))=\sum_{\{0,1}^n} P(x)*ln(P(x)), where n represents dimension. This will create vertices of sorts, use n=3 for example. We will get the following 8 vertices (0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), (1,1,1). If we group these terms based on the number of 1's in each vertex point, we get 1 term with zero 1's, 3 terms with one 1, 3 terms with two 1's, and one term with three 1's -> 1,3,3,1. If we consider the example with n=4, then we get 1,4,6,4,1, n=5 gives 1,5,10,10,5,1, and so on. This pattern is identical to pascals triangle. Also, all terms add up to (1/2^n), like for n=4: 1+4+6+4+1=16=(1/2)^4.
Then, another thing I noticed was the connection to the binomial distribution. If we calculate (n \\ k) meaning out of n choose k, for any n and k, we will get the values defined by the pascals triangle in the first paragraph. For example, with n=5:
(5 \\ 0) =5!/(5-0)!*0! = 1
(5 \\ 1) =5!/(5-1)!*1! = 5
(5 \\ 2) = 5!/(5-2)!*2! =10 and so on.
I want to check whether these relations have any validity or I am wasting my time with this. Any help here would be appreciated.