. Simplify each binomial coefficient or permutations to factorial fractions: n! n! k (n - k)!k! P(n, k) = - (n - k)! You need not simplify these expressions, such as 154 or 17!, in your response. Problem 1. 50 States (22 points) Suppose you are trying to rank the 50 states that make up the United States from best (1st) to worst (50th). (a) [2 pts.] How many different possible rankings are there of the 50 states? (b) [2 pts.] New England consists of six states (Maine, Vermont, New Hampshire, Massachusetts, Rhode Island, Connecticut). How many different ways could these states be ordered within the ranking? For example, one order would be Connecticut, then Massachusetts, then Vermont, then New Hampshire, then Maine, then Rhode Island. (c) [2 pts.] If we consider the positions occupied by the New England states, how many different sets of positions could there be? For example, one set could be where the New England states represent positions 11, 13, 20, 28, 31, and 33. (d) [2 pts.] Now, if we consider the positions occupied by the New England states and the states in those positions, how many different possibilities are there? For example, one possibility could be where Massachusetts is 11, New Hampshire is 13, Maine is 20, Vermont is 28, Connecticut is 31, and Rhode Island is 36. (e) [3 pts.] How many final rankings of the 50 states have one of the New England states in 10th place? Briefly describe how you arrived at your solution. (f) [3 pts.] How many final rankings of the 50 states have none of the New England states in 10th place? Briefly describe how you arrived at your solution. (g) [4 pts.] If we assume that none of the New England states are in the top 10, then how many final rankings of the 50 states are there? Describe how you arrived at your solution. (h) [4 pts.] How many final rankings of the 50 states have Vermont before Rhode Island before Maine before New Hampshire? One such example would be Vermont in 4th, Rhode Island in 8th, Maine in 19th, and New Hampshire in 30th. Describe how you arrived at your solution. Solution