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Show your thinking before showing the numbers! You must show your work completely and neatly for full credit. 1. I am adding to my standard 52-card deck two wild cards. (A "wild card" is a playing card that can have any value, suit, color, or other property in a game at the discretion of the player.) With this 54-card deck, how many 5-card hands give a "straight" (five cards with consecutive faces. The ace can be used either as the lowest or the highest face)? 2. In a department of Mathematics with 20 faculty and 8 admin members, a committee of 7 faculty and 3 admin must be formed. Also, two of the 7 chosen faculty members must be designated as chair and vice-chair of the committee. In how many ways can this be done? 3. An urn contains 12 marbles: 3 blue, 4 red, 5 green. In how many ways can I: (a) line up all of them; (b) line up five of them; (c) extract five of them at once, without order. 4. Give a combinatorial proof of the following formula: a, n b, C ) = 3". atbto-n (do not use the the Trinomial Formula!)

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