1) working out the number of combinations including duplicate scoops
Consider the case where there is only one scoop of ice cream. There are 16 flavors (choices), and thus 16 "combinations." The next case is 2 scoops. One way to think about this problem is to consider how many choices you have per scoop. There are 16 choices for the first scoop and 16 choices for the second scoop since duplicates are allowed. This works out to $16^2 = 256$. It should be more clear as to how you can expand this to more scoops.
2) working out the number of combinations where a flavour only appears
once in each possible combination (e.g. chocolate-vanilla-strawberry
and then vanilla-chocolate-strawberry)
Now try the case of 3 scoops. There are 16 choices for the first scoop, but 15 choices for the second scoop since duplicates are not allowed. For the third scoop there are 14 choices, which works out to $16*15*14=3,360$. Note that with this way of combining flavors, the order is important since chocolate-vanilla-strawberry and vanilla-chocolate-strawberry are both counted.
EDIT (for cases where duplicates are allowed):
1.If there are 16 different ice-cream flavours, how many combinations are there for a two scoop?
There's a decent explanation from another post. The formula is indeed
where $n$ is the number of flavors and $k$ is the number of scoops. This is called a combination. Note that using this formula gives a different answer than I originally provided, since combinations do not consider the order in which objects are counted
- If there are still 16 different ice-cream flavours, how many combinations are there for a three scoop?
Using the formula above it can be worked out similarly