Math Probability Data Management Problem?

Hey everyone, I received these two questions as an assignment due in a few days. Can anyone help me figure it out?





You have k different pairs of shoes, where k is an integer greater or equal to 1. how many ways can you arrange the shoes in a row so that no shoe is ever located beside its partner?


(Basically find the formula in terms of k)





Determine the number of ways that a random five-card hand (from a standard deck of cards) can be formed so that the hand has exactly one pair (a hand of three-of-a-kind does not qualify, a hand of two pairs does not qualify. so basically, there must be one pair and the other three cards must be different).





Thanks a lot guys. I tried making a giant tree to work out the actual probability, but it%26#039;s not working so well....anyone have any idea?

Math Probability Data Management Problem?
The second question is easier.





For any question involving cards, think of choosing a card as two choices - the suit and the rank.





(I%26#039;m going to use n C r notation to mean n choose r, or n combination r = n!/(r!(n-r)!)





So for a five card hand to have exactly one pair, you can choose the rank of the pair 13 C 1 = 13 ways.





You can choose their suits 4 C 2 = 6 ways.





Of the three remaining cards, you can choose their ranks 12 C 3 ways = 220 ways (12 because of the 13 ranks, one was used up in your pair.)





and each of those cards has 4 C 1 = 4 choices for its suit.





So there are 13 * 6 * 220 * 4 * 4 * 4 = 1098240 different hands with exactly one pair in them.





As for your first question, what you are looking for is the number of derangements. A derangement is an arrangement with every element out of order. I found the wikipedia page, and a planetmath page, but both give a recursive algorithm, not a formula per se.



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