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

Let a be the number of objects before first selected object, b and c are those between first and second and second and third respectively and d is that after third selected object. Then $aIbIIcIIIda+b+c+d=n-3$  where $a,d\ge 0, and b,c\ge 1.$  Number of ways = coeff. Of $x^{n-3}in{\left(1+x+x^2+\mathrm{.....}\right)}^2{\left(x+x^2+\mathrm{....}\right)}^2$  = coeff. Of $x^{n-5}in{\left(1+x+x^2+\mathrm{.....}\right)}^4$  $={}^{n-2}{C}_{n-5}={}^{n-2}{C}_3$