### From Coin Tossings to Combinations - Mark Holland

##### 25th January 2014

We will explore various problems in counting, and connections to coin tossing sequences and probability. To get you started, consider the following problems:Problem: How many distinct arrangements are there of the string of letters "ABCDE?" How about "AABBB?" How does this generalize?

Problem: I have 3 boxes, numbered 1,2 and 3. I have 5 identical balls to place in the boxes (empty boxes allowed). How many ways can I do this? How does this generalize?

Problem: If I toss a coin "n" times, how many sequences of "Heads" and "Tails" do I get? How many sequences do I get if I disallow sequences where we have a "Tail followed by a Tail"?

Problem: (Given time): Two players choose sequences they want to see appear first. Player 1 chooses sequence HH, while player 2 chooses sequence TH, where T=tail, H=Head. Whose sequence is more likely to appear first - or are they equally likely? If player 1 chooses a sequence of length three (e.g. THH), can player 2 always choose a sequence that's more likely to appear before the choice of Player 1?