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

Options

A. `.^(n-2)C_(3)`
B. `.^(n-3)C_(3)+.^(n-3)C_(2)`
C. `((n-2)(n-3)(n-4))/(6)`
D. `.^(n)C_(2)`

Answer

A, B, C

Explanation

Let `a_(0)` be the number of objects to the left of the first object chosen, `a_(1)` be the number of objects betweenn the first and the second `a_(2)` be the number of objects between the second and the third and `a_(3)` be the number of objects to the right of the third object. then,

also `a_(0)+a_(1)+a_(2)+a_(3)=n-3`
let `a=a_(0)+1,b=a_(3)+1,` then `age1,bge1` such that
`a+a_(1)+a_(2)+b=n-1`
The total number of positive integral solutions of this equation is `.^(n-1-1)C_(4-1)=.^(n-2)C_(3)+.^(n-3)C_(3)+.^(n-3)C_(2)`
`=((n-2)(n-3)(n-4))/(1*2*3)`