No 3's Allowed!

Let \(P = (p_1, p_2, \ldots, p_7)\) be a permutation of the integers \(1, 2, \ldots 7\). For how many permutations \(P\) are all seven sums \(S_1 = p_1\), \(S_2 = p_1 + p_2\), \(\ldots\) and \( S_7 = p_1 + p_2 + \cdots + p_7\) not multiples of 3?

Details and assumptions

A permutation is a rearrangement of the entire set of objects.

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