Use the Map operator. The shorthand symbol is /@. Note the use of the # symbols. These are placeholders for the Map function which maps the values from the set {1,2,3,4} into these places. The & symbol designates the Sum expression as a pure function with the list {1,2,3,4} mapped sequentially into the placeholder expressions.

Code:In[15]:= (Sum[(Binomial[12, i]*Binomial[40, #1 - i])/ Binomial[52, #1], {i, 1, #1}] & ) /@ {1, 2, 3, 4} Out[15]= {3/13, 7/17, 47/85, 2759/4165}