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1. (7 points) Please choose ONE of the following questions to answer. Please indicate clearly which question you are answering.
(i) You flip a coin 1000 times. Determine the probability of getting 600 heads and 400 tails. Express your answer in the form , where is some number and is some integer. You will likely need to use Stirling’s approximation, but you’ll need to use your best judgment as to which version is most appropriate. (Note that the math here is the same as the math we would do for a two-state system).

(ii) A sandwich aficionado (let’s call him Joey) walks to his favorite deli everyday. He begins at his coffee shop, at location C, and walks mblocks east and nblocks north to the deli, at location D. (The sketch to the right shows a possible path Joey can take in the case and ; note that he always stays “on the grid”, and can’t take any diagonal shortcuts). Since Joey is eager to get his sandwich, he always approaches the deli from the coffee shop (in other words he never doubles back). If the deli is actually 10 blocks east and 15 blocks north of the coffee shop, how many different ways are there for Joey to walk from the coffee shop to the deli (assuming he always stays on the grid and never doubles back)? (Note we could use the idea of this problem to model the motion of molecules from one location to another.)