If ³⁵ C _ x+7 = ³⁵ C _ 4 x-2 , then the values of x are either

Step-by-step solution

The property of combinations states that if ^ n C_ a = ^ n C_ b , then either a = b or a + b = n . Given ³⁵C_ x+7 = ³⁵C_ 4x-2 , we have two cases. Case 1: x + 7 = 4x - 2 . Rearranging the terms gives 3x = 9 , which implies x = 3 . Case 2: (x + 7) + (4x - 2) = 35 . Simplifying the expression gives 5x + 5 = 35 . Subtracting 5 from both sides gives 5x = 30 , which implies x = 6 . Thus, the possible values for x are 3 or 6. Answer: 3 or 6