The number of elements in the set S = (r, k) : k Z and 36 r+1 = 6 ( ³⁵C_r ) (k^2 - 3) , is :

Step-by-step solution

Given equation: ³⁶C_ r+1 = 6 (³⁵C_r ) k^2 - 3 Using the property ^ n C_r = n r n-1 r-1 , we can write: ³⁶C_ r+1 = 36 r+1 ³⁵C_r Substituting this into the given equation: 36 r+1 ³⁵C_r = 6 k^2 - 3 ³⁵C_r For ³⁵C_r to be defined, r must be an integer such that 0 r 35 . Thus, ³⁵C_r 0 . Dividing both sides by ³⁵C_r : 36 r+1 = 6 k^2 - 3 6 r+1 = 1 k^2 - 3 k^2 - 3 = r+1 6 k^2 = 3 + r+1 6 For k Z , k^2 must be a perfect square integer. This requires r+1 to be a multiple of 6 . Since 0 r 35 , we have 1 r+1 36 . Let r+1 = 6m ,