Question

For his birthday party, Gojo’s 3 students bought him a cake cut into 10 slices. Give your answer for both parts as an integer with NO COMMAS, NO SPACES, and NO LABELS.

How many ways are there to distribute the 10-slice cake among 4 people, if half of the cake is ice cream cake and the other is red velvet cake?

Answer 1

The total number of ways to distribute a 10-slice cake with two different flavors among four people is 3136.

Step-by-step explanation:

The question revolves around the combinatorial problem of distributing slices of cake, that are of two different types, among four people. To tackle this, we use the stars and bars theorem, a well-known combinatorial technique. We will break down the problem into distributing ice cream cake slices and red velvet cake slices separately, assuming there are 5 slices of each type.

Since there are 5 slices of ice cream cake to distribute among 4 people, the number of distribution ways can be calculated as the number of solutions to the equation x1 + x2 + x3 + x4 = 5, where xi is the number of slices the ith person gets. Using stars and bars, the number of ice cream distributions is combinations(5 + 4 - 1, 4 - 1), which is 56.

As the red velvet cake is also cut into 5 slices, we calculate in the same way, yielding another 56 ways. Since each distribution of ice cream cake slices is independent of the red velvet cake slices, we multiply the two numbers to get the total distribution ways for the whole cake, which is 3136.