Problem Session Problems
1. How many strings of length 4 can be made out of AABBCDE?
The most important part to focus on in this question is the different combinations of A and B we can have in our final string. In order to solve this question we break it down into 4 distinct cases:

Case 1: All letters distinct:
In this case we are really only selecting our letters from ABCDE, so we have 5*4*3*2 = 120 total combinations here. This covers both the case of having a single A, having a single B, and having both a single A and a B.

Case 2: Pair of A or pair of B
In this case we solve first for having a pair of A's, then multiply by 2 because the two cases are symmetric. We first choose 2 spots to put our A's in, which can be done in 4C2 ways. We then place the remaining 3 possible letters in the last 2 spots in 3*2 ways. Thus we have 2 * 4C2 * 3 * 2 = 72 total combinations.

Case 3: Pair of A and one B or pair of B and one A
Here we focus on first placing our single letter that is not an A or B. We have 3 letters to choose from, C, D, or E, and 4 places, for a total of 3*4. Of the three remaining spots, we can arrange a pair of A's and one B 3 ways, for a total of 2 * 3 * 4 * 3 = 72 total combinations.

Case 4: Pair of A and pair of B
In this case we only need to place our pair of A's down, and the B's have to fill in the remaining spots. We can choose the spots for the two A's in 4C2 = 6 total ways.

Adding up all our cases we get 120 + 72 + 72 + 6 = 270 total strings of length 4.

2. How many strings of length 11 can be made out of MISSISSIPPI?
We have 4 I's, 4 S's, and 2 P's. The total number of strings is then 11!/(4!4!2!)

BONUS: How many Hamiltonian paths are there from one corner of a cube to the other, traveling only along the edges of the cube?