1. Prove that the probability of exactly n heads in 2n tosses of a fair coin is given by the product of the odd numbers up to 2n - 1 divided by the product of the even numbers up to 2n.

2. Let n be a positive integer, and assume that j is a positive integer not exceed- ing n/2. Show that in Theorem 3.5, if one alternates the multiplications and divisions, then all of the intermediate values in the calculation are integers. Show also that none of these intermediate values exceed the final value.