1. (20%) Solve $a_n = 3a_{n-1}^2, \, a_0 = 1.$

2. (20%) Consider the permutations of 1, 2, 3, 4. The permutation 1432 is said to have one ascent (since 1 < 4) and two descents (since 4 > 3 and 3 > 2). Suppose a permutation of 1, 2, 3,…, m has k ascents, for $0 \leq k \leq m-1.$ How many descents does the permutation have?

3. (10%) Given 8 Perl books, 17 Python books, 6 Java books, 12 SQL books, and 20 Objective-C books, how many of these books must we select to insure that we have 10 books dealing with the same computer language?