On selecting \(k\) balls from an \(n\)-line without unit separation (Q1096632)

Let \(\ell(n,k)\) denote the number of ways of selecting \(k\) balls from \(n\) balls arranged in a line (called an \(n\)-line) with no two adjacent balls, from the \(k\)-selected balls, being unit separation. It is shown that \(\ell(n,k)\) satisfies the recurrence relation \(\ell(n,k)=\ell (n-1,k)+\ell(n-1,k-1)-\ell(n-2,k-1)+\ell(n-3,k-1)\) and an explicit form for \(\ell(n,k)\) is obtained.