Question

A bake shop, offers many variety of cakes to customers according to the following:

1. The customer has to pick one of these cake sponge: white cake, chocolate, or red velvet.

2. It is possible for a customer to have any combination of fillings such as cream cheese, vanilla buttercream, chocolate ganache, strawberry, chocolate buttercream, mango filling, and mocha buttercream.

3.The customer may add sprinkles, chocolate chips, brownies, and/or sugar candy. These additional toppings are optional.

Assume that a cake is required to have a cake sponge and at least one type of filling, how many varieties of cakes are offered in this bakery?

Answer 1

1) A sponge say choclate may go with 7 different types of fillings.. So 7 varieties of choclate cake.. Similarly other two sponges, each go with 7 varieties of filling hence all the possible combinations of sponge and fillings will results in 7*3=21 varieties...

2) But the assumption here is atleast one filling should be used. In that case combination of fillings need to be considered.. Combination of two fillings happen in 7C₂ ways.. Similarly the combination of 3,4,5,6,7 fillings happen in 7C₃,7C₄,7C₅,7C₆ and 1 ways respectively. Hence the sum these combinations = 127 possible combinations. There for one sponge may go with these 127 combinations... Therefore the total varities of cake is 3*127=381 varieties. Please ignore the second part if that is not you are looking for...