For a fixed integer $$n>1$$ , let $$\mathscr {B}_n=\{ \pm a_1, \pm a_2, \pm a_3, \dots , \pm a_{n-1}, a_n \}$$ , where $$a_i \in \mathbb {R}^+$$ , $$i= 1,2, 3, \dots , n$$ , and $$a_1 < a_2 < a_3 < \dots < a_n$$ and $$-a_n \notin \mathscr {B}_n$$ . Let $$\phi (\mathscr {B}_n)$$ be the set of all non-empty subsets of $$\mathscr {B}_n$$ satisfying the condition $$a_1 < a_2 < a_3 < \dots < a_n$$ and $$-a_n \notin A$$ for each $$A \subset \mathscr {B}_n$$ . Let $$\mathscr {B}_n^+=\{ a_1, a_2, a_3, \dots , a_{n-1}, a_n \}$$ and $$V_1$$ be the set of k-element subsets of $$\mathscr {B}_n^+$$ , $$1 \leq k