Question 27
Let $n$ be a positive integer. We are given circular disks with radii $1,2, \ldots, n$. Of each size we have two disks: a transparent one and an opaque one. Each disk has a small hole exactly in its centre, with which we can stack the disks on a vertical rod. We now want to make stacks satisfying the following conditions:
- Of each size exactly one disk lies on the stack.
- Looking straight down from above, we can see the outer edges of all $n$ disks on the stack. (That is, if an opaque disk lies on the stack, then no smaller disks may lie below it.)
Determine the number of different stacks we can make under these conditions. (Two stacks are different if they do not use exactly the same set of disks, but also if they do use exactly the same set of disks but not in the same order.)
- Of each size exactly one disk lies on the stack.
- Looking straight down from above, we can see the outer edges of all $n$ disks on the stack. (That is, if an opaque disk lies on the stack, then no smaller disks may lie below it.)
Determine the number of different stacks we can make under these conditions. (Two stacks are different if they do not use exactly the same set of disks, but also if they do use exactly the same set of disks but not in the same order.)