A finite set of vectors 5 subset of R(n) is called a simplex if S is linearly dependent but all its proper subsets are independent. If we denote the number of simplexes contained in H subset of or equal to R(n) by simp(H), then our main results can be formulated as: THEOREM: For any H subset of or equal to R(n) of fixed size, simp(H) is maximal if any n vectors H are linearly independent. THEOREM: For any H subset of or equal to R(n) of fixed size so that H spans R(n), simp(H) is minimal if H consists of n collections of parallel vectors of sizes differing by at most one from each other. COROLLARY: Let H subset of or equal to R(n) so that H spans R(n) and \H\ = m. Then writing m = an + b where 0 less than or equal to b < n, we have: [GRAPHICS] The related problem regarding the minimum value of simp(H) under the condition that parallel vectors are not allowed in H remains open.