Question

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

(A)  (n - 2)(n - 3)(n - 4)/6

(B)   ^n–2C₃

(C)   _^n–3C₃ +  ^n–3C₂

(D)   All of these

Answer 1

Correct option (D)  All of these

Let x₀ be the number of objects to the left of the first object chosen, x₁ the number of objects between the first and the second, x₂ the number of objects between the second and the third and x₃ the number of objects to the right of the third object. We have

x₀ , x₃ ≥ 0 ,x₁ , x₂ ≥ and x₀ + x₁ + x₂ + x₃ = n – 3 ......(1)

The number of solutions of Eq. (1)

= Coefficient of y^n–3 in (1 + y + y² + …)(1 + y + y² + …)(y + y² + y³ + …)

= Coefficient of y^n–3 in y² (1 + y + y² + y³ + …) ⁴ 

= Coefficient of y^n–5 in (1 – y) ^–4

= Coefficient of y^n–5 in (1 + ⁴C₁ y + ⁵C₂ y₂ + ⁶C₃ y³ + …)