ILL 1992 Problem 41

Let $S$ be a set of positive integers $n_1, n_2, \cdots, n_6$ and let $n(f)$ denote the number $n_1n_{f(1)} +n_2n_{f(2)} +\cdots+n_6n_{f(6)}$ , where $f$ is a permutation of $\{1, 2, . . . , 6\}$ . Let $$\Omega=\{n(f) | f \text{ is a permutation of } \{1, 2, . . . , 6\} \}$$ Give an example of positive integers $n_1, \cdots, n_6$ such that $\Omega$ contains as many elements as possible and determine the number of elements of $\Omega$ .