Combination/Permutation: number of n-scoop ice cream cones

[WTF?]

Jan at a 31 flavors Baskin Robins. Jan visits a Baskin Robins store that sell 31 flavors of ice cream.

1)How many different 1 scoop ice cream cones could Jan order?
2)How many different 2 scoop ice cream cones could Jan order?
3)How many different 3 scoop ice cream cones could Jan order?
4)How many different 4 scoop ice cream cones could Jan order?
5)How many different n scoop ice cream cones could Jan order?
6)How many different 1 scoop ice cream cones could Jan order if she doesn’t repeat any flavors?
7)How many different 2 scoop ice cream cones could Jan order if she doesn’t repeat any flavors?
8)How many different 3 scoop ice cream cones could Jan order if she doesn’t repeat any flavors?
9)How many different 4 scoop ice cream cones could Jan order if she doesn’t repeat any flavors?
10)How many different n scoop ice cream cones could Jan order if she doesn’t repeat any flavors?

OK, so basically, I can recognize the pattern I think, basically the first couple of questions are just factorials? The longer questions are permutations, right?

If so, how do I justify my answers without just stating or recognizing the pattern? Can someone help on one example, please? How would I justify, say, example 5 and example 9?

 

[pka]

N is the number of scoops.

For 1-5 \(\displaystyle {N+30}\choose N\).

For the rest \(\displaystyle {31}\choose N\).