Let i represent the index (num in your example) and
    j represent the actual word
    c represent all the letters ('A'..'Z')

Let X[i,j] = { 1 if word (i,j) was chosen, 0 otherwise}

The value of each word can be broken into characters/letters

For all (i, j) - 
  # you get the value only if the word is chosen
  V[i,j,c] = number of c's in the word (i,j) * X[i,j]   

For all c
  Let Value(c) = (sum {over all i,j} V[i,j,c])*((sum {over all i,j} V[i,j,c])+1)/2  

Optimization problem
===================
Max sum {over c} Value(c)

# You can choose only one word from each index (num)
subject to: sum {over all j} X[i,j] = 1 for all i.
subject to: x[i,j] element of (0,1) for all i, j.