Permutations and Combinations

Order counts in permutations. For combinations, only the final result matters, not how you get to it. Order does not count for combinations.

This is asking for combinations with 5 items (types of toppings) taken 2 at a time. $5!/(3! \times 2!) =120/12 = 10$.

 Since these are independent events you apply the multiplication rule: (.5 x .1667 = .0833)

This is asking for permutations with 5 items (entries) taken 2 at a time (first place and second place). $5!/(3!) =120/6 = 20$.