Consider the set S= (a, b, c, d, e): 0 < a < b < c < d < e < 100 where a, b, c, d, e are integers. If D is the average value of the fourth element of such a tuple in the set, taken over all the elements of S , find the largest integer less than or equal to D .

Step-by-step solution

array ll d=4 & sum =4 ⁹⁵ C₁ ^3 C₃ d=5 & sum =5 ⁹⁴ C₁ ^4 C₃ d=6 & sum =6 ⁹³ C₁ ^5 C₃ & d=98 & sum =98 ^n C₁ ⁹⁷ C₃ array Total =4 ⁹⁵ C₁ ^3 C₃+5 ⁹⁴ C₁ ^4 C₃ 98 ^1 C₁ ⁹⁷ C₃ aligned & =4 _ r=1 ⁹⁵ (99-r) 4 r^ 98-r C₃ & =4 _ r=1 ⁹⁵ ^ 99-r C₄ r & =4 ¹⁰⁰ C₆ & A.M = 4 ¹⁰⁰ C₆ ⁹⁹ C₅ & = 200 3 & [ 200 3 ]=66 aligned