The poset Y-k,Y-2 consists of k + 2 distinct elements x(1), x(2), ..., x(k), y(1), y(2), such that x(1) <= x(2) <= ... <= x(k) <= y(1), y(2). The poset Y-k,Y-2' is the dual poset of Y-k,Y-2. The sum of the k largest binomial coefficients of order n is denoted by Sigma(n, k). Let La-#(n, {Y-k,Y-2, Y-k,Y-2') be the size of the largest family F subset of 2([n]) that contains neither Y-k(,2) nor Y-k,Y-2' as an induced subposet. Methuku and Tompkins proved that La-#(n,{Y-2(,2), Y-2,Y-2'}) = Sigma(n, 2) for n >= 3 and conjectured the generalization that if k >= 2 is an integer and n >= k +1, then La-#(n, {Y-k,Y-2, Y-k,Y-2'}) = Sigma(n, k)) . On the other hand, it is known that La-#(n, Y-k,Y-2) and La-#(n, Y-k,Y-2') are both strictly greater than Sigma(n, k). In this paper, we introduce a simple approach, motivated by discharging, to prove this conjecture.