For integers n and r, let  nr= nCr,if    n≥r≥00,otherwise                 The maximum value of k for which the sum ∑i=0k+110i15k−i+∑i=0k+112i13k+1−i exists, is equal to

∑i=0kCi   10·Ck-i   15= coeff. of xk in (1+x)25=Ck   25∑i=0k+1Ci   12Ck+1-i   13= coeff. of xk+1 in (1+x)25=Ck+1   25∣Sum is Ck   25+Ck+1   25=Ck+1   26Max. value of ‘k’ for which 26Ck+1  exists is k = 25