Collect columns in a table

Question

How can I add the elements of certain rows and show them as a new row in the table?

The following table shows the required process:

My attempt is just to make the table

A = Tuples[{0, 1}, 4]

I'm having trouble creating a new column that represents the sum applied in general

Any Help

Answer 1

Clear["Global`*"]

SeedRandom[1234];

(data = RandomChoice[{0, 1}, {16, 4}]) // MatrixForm

(table = Prepend[
    Transpose@
     Join[{Range@Length@data}, 
      Transpose@
       data, {data[[All, 1]] + data[[All, 3]]}, {data[[All, 2]] + 
        data[[All, 4]]}],
    Flatten[{"order", Subscript["x", #] & /@ Range[4], 
      StringForm["``=``+``", Subscript["l", #[[1]]], 
         Subscript["x", #[[2]]], Subscript["x", #[[3]]]] & /@
       {{1, 1, 3}, {2, 2, 4}}}]]) // Grid[#, Frame -> All] &