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Unformatted text preview: Math 55 Second Midterm 7 Nov 2000 NAME (1 pt): TA & section #(1 pt): Name of Neighbor to your left (1 pt): Name of Neighbor to your right (1 pt): Instructions : This is a closed book, closed notes, closed calculator, closed computer, closed network, open brain exam. You get one point each for filling in the 4 lines at the top of this page. All other questions are worth 10 points. Since all questions have equal weight, do the questions you find easiest first. If you start taking this exam, you have to turn it in. Write all your answers on this exam. If you need scratch paper, ask for it, write your name on each sheet, and attach it when you turn it in (we have a stapler). 1 2 3 4 5 Total 1 Question 1) (10 points) From a group of 2 freshman (Sam and Pam), 3 sophomores (Will, Bill, and Jill) and 4 professors (Jones, Stones, Clones, and Bones) we wish to arrange a picture of 5 of them standing in a row. How many ways are there to arrange the 5 people in a photograph under the following conditions (each set of conditions is independent of the others). You may leave expressions like C(20,10), or 13! unsimplified in your answers. 1. No conditions: any 5 people may appear in any order Answer: P(2+3+4,5) = P(9,5) = 9!/4! = 15120 2. There must be exactly 3 faculty in the photograph. Answer: (# ways to pick 3 locations for a professor) * (# ways to arrange 3 professors in order) * (# ways to fill remaining 2 slots) = C(5,3)*P(4,3)*P(2+3,2) = 4800 3. A freshman must appear directly to the right of a sophomore, and that sophomore directly to the right of a professor in the picture. Answer: (# places to start sequence freshman/sophomore/professor) * (# ways to pick the freshman) * (# ways to pick the sophomore) * (# ways to pick the professor) * (# ways to fill in remaining 2 slots) = 3 * 2 * 3 * 4 * P(2+3+43,2) = 2160 2 Question 1) (10 points) From a group of 3 boys (Joe, Bill and Lou), 3 girls (Zoe, Jill and Sue) and 4 school teachers (Dingle, Pringle, Zingle, and Single) we wish to arrange a picture of 5 of them standing in a row. How many ways are there to arrange the 5 people in a photograph under the following conditions (each set of conditions is independent of the others). You may leave expressions like C(20,10), or 13! unsimplified in your answers. 1. No conditions: any 5 people may appear in any order. Answer: P(3+3+4,5) = P(10,5) = 10!/5! = 30240 2. There must be exactly 2 school teachers in the photograph. Answer: (# ways to pick 2 locations for a teacher) * (# ways to arrange 2 teachers in order) * (# ways to fill remaining 3 slots) = C(5,2)*P(4,2)*P(3+3,3) = 14400 3. A boy must appear directly to the right of a girl, and that girl directly to the right of a teacher in the picture....
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 Spring '08
 STRAIN
 Math, Number Theory, Metropolitana di Napoli, Prime number, A Closed Book

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