An ice cream shop offers a choice of three types of cones and 31 flavors of ice cream. A customer gets to choose a cone and a type of ice cream. How many options can customer choose?

a) each flavor must be different and the order of flavors is unimportant?

31!/3! (28) !

Yes. 31C3 or (313) counts the ways to select 3 unique items from 31.

b) each flavor must be different and the order of flavors is important?

31! / (28) !

Likewise, 31P3 or (313) 3! is the ways to select 3 from 31 and arrange them.

c) Flavors need not be different and the order of flavors is unimportant? (This is a non-trivial question)

33!/3! (30) !

Indeed! The "stars and bars" method counts (31+3-131-1) ways to put 3 identical items into 31 distinct boxes - or in this case take 3 scoops from 31 tubs.

Alternatively you might have counted the ways to select: three identical scoops, or a pair and a single, or three different scoops. 31+31⋅30 + (313)

d) Flavors need not be different and the order of flavors is important?

31∗31∗31

Yes, 313 counts the ways to make 3 independent choices with 31 options each