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Math 413 Homework 3

In order to obtain full credit, your solution should be clearly written with full sentences and explanation of your steps. Each problem below is worth 5 points. Problems 2-4 are from chapter 2 of the 5th edition.

1. You are an amazingly successful hotelier; one of your properties is "Chez Four-Thirteen". Suppose there are 8 distinguishable hotel rooms (numbered 101, 102,...,108), each with maximum occupancy of 4 people. A bus comes in with anywhere from 1 to 32 (distinguishable) people, but you do not know exactly how many. While waiting, you decide to calculate (making use of your UIUC Math 413 education) the number of possible ways the people on the bus can spend the night in those hotel rooms so you can make a welcome sign. Determine this number. Use a computer to express the answer with all the digits (e.g., 31415926535897932384626433832795); write down for your check-in clerk the order of magnitude (e.g., 1031). I recommend python to do the computation.

2. (Q57) What is the probability that a poker hand contains exactly one pair (that is, a poker hand with exactly four different ranks?)

3. (Q60) A bagel store sells six different kinds of bagels. Suppose you choose 15 bagels at random. What is the probability that your choice contains at least one bagel of each kind? If one of the bagels types is Sesame, what is the probability that your choice contains at least three Sesame bagels AND AT LEAST ONE BAGEL OF EACH KIND?

4. (Q64) Let n be a positive integer. Suppose we choose a sequence i1,i2,...,in of integers between 1 and n at random. What is the probability that the sequence contains exactly n-2 different integers?

5. Suppose n people line up to get into q different clubs. How many ways are there to do it? (The people are distinguishable and the order people are in line matters.)