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1. \(2^8 = 256\) choices. You have two choices for each tie: wear it or don’t.

2. You have 7 choices for regular ties (the 8 choices less the “no regular tie” option) and 31 choices for bow ties (32 total minus the “no bow tie” option). Thus total you have \(7 \cdot 31 = 217\) choices.

3. \({3\choose 2}{5\choose 3} = 30\) choices.

4. Select one of the 3 bow ties to go on top. There are then 4 choices for the next tie, 3 for the tie after that, and so on. Thus \(3\cdot 4! = 72\) choices.