There are $32$ ways to place the first object. After that there are $28$ places for the second object. That is, there are $32\cdot28=896$ ways to place the first two objects.
If the first two objects are $2$ places apart, then there are $25$ places for the third; if the first two places are $15$ apart, there are $26$ places for the third; otherwise, there are $24$ places.
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Of the $896$ ways to place the first two, $64$ have them being two apart and $64$ have them being $15$ apart. Thus, there are $$ \overbrace{\ 64\cdot26\ }^{\text{first two $15$ apart}}+\overbrace{\ 64\cdot25\ }^{\text{first two $2$ apart}}+\overbrace{768\cdot24}^{\text{otherwise}}=21696 $$ ways to pick three objects where we care about their order. For each of the ways to arrange without caring about order, there are $3!=6$ ways to reorder the objects; therefore, if we don't care about the order, there are $$ \frac16\cdot21696=3616 $$ ways to arrange three objects.