If n objects are arranged in a row, then the number of ways of selecting three objects such that no two of them are next to each other is

Step-by-step solution

Let (x ) be the number of objects to the left of the first object chosen, (y ) the number of objects between the first and the second, ( z ) the number of objects between the second and the third and (u ) the number of objects to the right of the third objects. Then, (x, u 0 ; y, z 1 ) and (x+y+z+u=n-3 ). Let (y₁= ) ( y -1 ) and ( z ₁= z -1 ). Then, ( y ₁ 0, z ₁ 0 ) such that ( x + y ₁+ z ₁+ u = n -5 ). The total number of non-negative integral solutions of this equation is ( ^ n -5+4-1 C _ 4-1 = ^ n -2 C ₃ . )