In a T20 cricket match, three players X, Y and Z made 37 runs in total. The ratio of the number of runs made by X to the number of runs made by Y equals the ratio of the number of runs made by Y to the number of runs made by Z. Quantity-I = Runs made by X Quantity-II = Runs made by Y Quantity-III = Runs made by Z Which one of the following is correct?

Given X/Y = Y/Z, so Y² = XZ, and X + Y + Z = 37. We need integer solutions (runs are non-negative integers). Y² = XZ with X+Y+Z = 37. Try Y=12: Y²=144, need XZ=144 and X+Z=25; solving t² − 25t + 144 = 0, discriminant = 625−576 = 49, t = (25±7)/2 = 16 or 9. So X=16, Z=9 or X=9, Z=16. Both satisfy. In one case X>Y>Z, in the other X<Y<Z. So we can't determine the ordering uniquely. Try Y=6: 36=XZ, X+Z=31, t²−31t+36=0, discriminant=961−144=817, not a perfect square. Try Y=10: 100=XZ, X+Z=27, t²−27t+100=0, discriminant=729−400=329, not a perfect square. So multiple valid orderings exist (16,12,9 and 9,12,16). Hence the ordering cannot be determined uniquely. Option (d) is correct.