6. Caitlin is making a massive ice cream sundae to share with her friends. Suppose that she has 8
different flavors of chocolate-based ice cream and 9 different flavors of vanilla-based ice cream.
(a) (3 points) Assuming that Caitlin won't use the same flavor twice, in how many ways can
she make an ice cream sundae that uses 4 flavors of chocolate-based ice cream AND 3
flavors of vanilla-based ice cream?
$$C(8,4) \cdot C(9,3) = \frac{8!}{(8-4)!4!} \cdot \frac{9!}{(9-3)!3!} = \frac{8\cdot7\cdot6\cdot5\cdot4}{4\cdot3\cdot2\cdot1} \cdot \frac{9\cdot8\cdot7\cdot6\cdot5\cdot4}{6\cdot5\cdot4\cdot3\cdot2\cdot1} = \frac{1680}{24} \cdot 70 \cdot 84 = 5880$$
(b) (2 points) Assuming that Caitlin won't use the same flavor twice, in how many ways can
she make an ice cream sundae that uses 4 flavors of chocolate-based ice cream OR 3 flavors
of vanilla-based ice cream?
(c) (2 points) Assuming that Caitlin won't use the same flavor twice, in how many ways can
she make an ice cream sundae that uses 7 different flavors of ice cream?
(d) (3 points) Assuming that Caitlin won't use the same flavor twice, in how many ways can
she make an ice cream sundae that uses 7 different flavors of ice cream and uses only
chocolate-based ice cream or only vanilla-based ice cream?