Colored blocks and towers

Question

George got a big box from his parents. In this box are colored blocks. He has white, black, red, blue and orange blocks. These blocks are all exactly the same size and of he has the same amount of blocks for each color. George will build towers of $10$ blocks high. Two towers are equal if they have the same color on every level of the tower. How many different towers can George build?

A. Without white blocks?

B. With exactly $6$ white blocks?

C. With two blocks of each color?

A. Is this just $4^{10}$?

B. Is this ${{10}\choose{6}} + 4^4$?

C. Is this $5!$ ?

Answer 1

For C), we choose the $2$ positions (from the $10$) that will be occupied by colour 1. Then we choose the $2$ positions, from the remaining $8$, that are occupied by colour $2$. And so on.

For B), for every way of selecting where the whites will go, there are $4^4$ ways to fill in the rest. So you need to multiply, not add.

The answer to A) is correct.