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

Step-by-step explanation:

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