Which of the following is the greatest?

Question

Which of the following is the greatest? (A)  31 C02- 31 C12+ 31 C22-⋅s- 31 C312 (B)  32 C0 2- 32 C12+ 32 C22-⋅s+ 32 C322 (C)  32 C02+ 32 C12+ 32 C22-⋅s+ 32 C322 (D)  34 C02- 34 C12+ 34 C22-⋅s+ 34 C322

Tardigrade Admin · Accepted Answer

We know that n C02+ n C1 2+⋅s+ n Cn 2= 2 n Cn and n C02- n C12+⋅s+ n Cn2= = begincases 0, textif n is odd [2ex] n Cn / 2(-1)n / 2 textif n is even endcases From these , 31 C02- 31 C12+ 31 C22-⋅s- 31 C31 2=0 32 C02- 32 C12+ 32 C22-⋅s+ 32 C322= 32 C16 34 C02- 34 C12+ 34 C22-⋅s+ 34 C322=- 34 C17 32 C02+ 32 C12+ 32 C22-⋅s+ 32 C32 2= 64 C32 Obviously 64 C32, is greatest.

Q. Which of the following is the greatest?

$^{31} C_{0}^{2} -^{31} C_{1}^{2} +^{31} C_{2}^{2} - \dots -^{31} C_{31}^{2}$

$^{32} C_{0}^{2} -^{32} C_{1}^{2} +^{32} C_{2}^{2} - \dots +^{32} C_{32}^{2}$

$^{32} C_{0}^{2} +^{32} C_{1}^{2} +^{32} C_{2}^{2} - \dots +^{32} C_{32}^{2}$

$^{34} C_{0}^{2} -^{34} C_{1}^{2} +^{34} C_{2}^{2} - \dots +^{34} C_{32}^{2}$

Q. Which of the following is the greatest?

31C02​−31C12​+31C22​−⋯−31C312​

32C0​2−32C12​+32C22​−⋯+32C322​

32C02​+32C12​+32C22​−⋯+32C322​

34C02​−34C12​+34C22​−⋯+34C322​

$^{31} C_{0}^{2} -^{31} C_{1}^{2} +^{31} C_{2}^{2} - \dots -^{31} C_{31}^{2}$

$^{32} C_{0}^{2} -^{32} C_{1}^{2} +^{32} C_{2}^{2} - \dots +^{32} C_{32}^{2}$

$^{32} C_{0}^{2} +^{32} C_{1}^{2} +^{32} C_{2}^{2} - \dots +^{32} C_{32}^{2}$

$^{34} C_{0}^{2} -^{34} C_{1}^{2} +^{34} C_{2}^{2} - \dots +^{34} C_{32}^{2}$