For example, if n = 5 and k = 3, you might see something like this:

1 -> 3

4 -> 1

5 -> 2

(left -> right: place "right" somewhere on the right side of "left")

In this case, you have 10 possible options to place 5 cards following the 3 given orderings.

One such option is 4 -> 1 -> 5 -> 3 -> 2.

The machine picks each number completely randomly, so you may get duplicate orderings or reversed orderings (you have no options for this case), but not both number being the same (such as 1 -> 1).

Define E(n,k) as the expected number of possible arrangements of n cards, following k random orderings produced by the machine.

For example, E(2,1) = 1, E(3,1) = 3, E(3,2) =

**3/2**(this can be verified by brute force by hand).

Give the values of E(5,3) and E(10,5).

**Edit**: I made a mistake counting the number of ones in my hand-drawn table. Oops.