Remember: the number of permutations of n objects selected r at a time is n!/(n-r)!. the order in which things are chosen is significant.
The number of combinations is less: it's n!/((n-r)!r!). The order in which things are chosen is not significant.

1) There are 6 floats, and 3 choices are to be made among them. The order of selection matters, because the prizes are not the same: if floats A, B, and C win prizes, it matters whether A gets first prize, or B gets it, or C gets it.
So, use the permutation formula. n = 6 and r = 3, so the number of ways is 6!/(6-3)! = 720/6 = 120.

2) The order of selection does not matter here, since Nate wants to do both activities without having them be in any particular order.
So, use the combination formula. n = 7 and r = 2, so the number of ways is 7!/((7-2)!2!) = 5040/(120·2) = 21.

3) The order of selection does not matter here, since Gregory just wants to get 3 different boxes; it does not matter which one he picks up first.
So, use the combination formula. n = 5 and r = 3, so the number of ways is 5!/((5-3)!3!) = 120/(2·6) = 10.

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