17) Alberto has a large number of weights of 1, 3, and 9 grams. Wanting to use them to balance a 16-gram chain by placing them on only one of the two plates of a balance with equal arms, in how many different ways can he choose the weights?
$\begin{array}{ll}\text { (A) } 1 & \text { (B) } 3\end{array}$
(C) 7
(D) 9
(E) 16.
17) The answer is (D). If one decides to use a 9-gram weight, 3.2 of them can be used, or 0 of 3 grams (and, correspondingly, 1, 4, or 7 of 1 gram). If one decides not to use the 9-gram weight, 5, 4, 3, 2, 1, or 0 of 3 grams can be used (and correspondingly 1, 4, 7, 10, 13, 16 of 1 gram). In total, there are $3+6=9$ ways to choose the samples.