Based on the layer cake found at www.starchefs.com/recipes/layer-cake Pastry Chef Erin Kanagy-Loux has created a recipe for a layer cake that consists of five different kinds of cakes, and four different kinds of frostings, listed below. For simplicity, we will consider "cake" anything that is baked, and "frosting" anything that is spread. Cake flavors: Chocolate, Olive Oil, White, Buckwheat Chiffon, Dacquoise Frostings: Italian Buttercream, Mascarpone Crmeaux, Kefir Chocolate Ganache, Fermented Basmati Rice Custard. The recipe provides instructions for making each of the cakes and frostings. It concludes with the following instructions: Stack and layer cake as you please, until you're standing on your tippy-toes. Shape at will; be daring. Top with a healthy layer of Fermented Basmati Rice Custard. Steal the show. You are very opinionated about cakes and have decided that all layer cakes must begin with cake on the bottom; must alternate between cake and frosting; and must consist of ten layers of cake (and therefore ten layers of frosting). Note that the Chef instructs that the final layer of frosting be Fermented Basmati Rice Custard (FBR Custard). Yes Chef! (a) Your aunt wants to make the cake, but is uncomfortable with the complete lack of structure in the recipe. She has decided that her cake will use exactly two layers of each cake flavor, and three layers of each of the first three frostings (FBR Custard will only be used as the final layer). She will "go wild" by assembling the layers randomly. How many distinguishable cakes can result? Note: swapping two layers of the same flavor does not result in a distinguishable cake. (b) You aren't so rigid as your aunt and are willing to consider any distribution of the layers of cake and frosting- even if all 10 layers are the same flavor. You don't really care about the order of the layers, you only care about how many of each flavor are present in the cake (so, for example, if you and I make cakes consisting of precisely the same layers assembled in different orders, you will consider them the same cake). How many different cakes are possible? Keep in mind that the top layer must always be FBR Custard! (e) Your uncle argues that the way a cake is built absolutely affects the way it tastes. After all, no one can taste all ten layers at once, so the order of assembly absolutely affects the way it is experienced. He will design the cake by deliberately choosing a flavor for each layer of cake and for each layer of frosting. How many different cakes are possible? Again, remember the top layer restriction!