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 TOPICS This page:   permutations/combinations    Search   Dr. Math See also the Dr. Math FAQ:   permutations and   combinations Internet Library:   permutations/combinations HIGH SCHOOL About Math Analysis Algebra    basic algebra    equations/graphs/      translations    linear algebra    linear equations    polynomials Calculus Complex Numbers Calculators/    Computers Definitions Discrete Math    permutations/    combinations Exponents    Logarithms Fibonacci Sequence/   Golden Ratio Fractals Functions Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean    practical geometry    symmetry/tessellations History/Biography Interest Logic Negative Numbers Number Theory Physics/Chemistry Probability Projects Puzzles Sequences/Series Sets Square/Cube Roots Statistics Transcendental   Numbers Trigonometry Browse High School Permutations/Combinations Stars indicate particularly interesting answers or good places to begin browsing. Polya's Counting Theory [09/12/2003] In how many different ways could a 6-sided die be numbered? Polynomial Expansion [02/06/2003] What's the general formula for things like (a+b+c)^2; (a+b+c+d)^3; (a+ b+c+d+e)^4; (...n+1 terms...)^nth power? Possible Combinations [4/17/1996] My teacher found a master lock. On the lock there were 4 sections which held 10 numbers, 0 - 9. He asked us to tell him how many different combinations were possible. Possible Combinations from Cards Drawn at the Same Time? [04/02/1998] If two of four numbered cards are drawn at the same time, how many possible combinations could be made? Possible Combinations on a Push Lock [4/17/1996] A combination lock consist of a sequence of "pushes" and each push is a collection of buttons (1 or more) that are pushed together. Possible Combinations when Distributing Items Unevenly [05/24/2004] There are 20 distinct toys to be handed out to 5 children. How many possible ways are there to hand them out if two children each get 7 toys and three children get 2 toys each? Possible Paths across a Rectangular Grid [02/16/2005] Consider a grid that has 3 rows of 4 squares in each row with the lower left corner named A and upper right corner named B. Suppose that starting at point A you can go one step up or one step to the right at each move. This is continued until the point B is reached. How many different paths from A to B are possible? Probabilities of Picking Colored Balls out of an Urn [03/13/2006] An urn contains 8 white, 6 blue, and 9 red balls. How many ways can 6 balls be selected to meet various given conditons? Probability and Permutations [09/12/1999] A permutation f is a 1-1 mapping of the first n positive integers onto themselves. What is the probability that the permutation has the property that f(i) = i for at least one value of i, i between 1 and n inclusive? Probability of Duplicate Pairs [05/13/2003] Find the probability of getting at least 20 duplicate addresses when drawing a sample of 30,000 at random from UK households (estimate 21,000,000) where there is replacement every time a selection is made. Probability of Never Having a Losing Record [04/07/2001] Suppose a football team plays 8 games, and the chance of winning any particular game is 50%. What is the probability of completing the season without more losses than wins? Probability: Permutations and Combinations [12/10/2002] How is probability related to permutations and combinations? Product of Disjoint Cycles [10/16/1998] How to express (1 2 3 5 7)(2 4 7 6) as the product of disjoint cycles. Proof by Induction [4/3/1996] I was given a proof by my math teacher: by mathemetical induction, prove that i(nCi) = n2^n-1. Proof of Ordered Partioning of Integers [07/31/2001] I have found that there are 2^(n-1) ways to partition an integer (where order matters and all positive integers are available), but need a proof for this seemingly simple formula. Proof of the Addition Principle by Induction [07/18/1999] How can I prove by induction that (2^(3n)-1) is divisible by 7, for all values of n greater than 0? Puzzle to Find a 10 Digit Number [10/10/2004] I have to find a 10 digit number which uses each of the digits 0-9 such that the first digit is divisible by 1, the first two digits make a number divisible by 2, the first three digits make a number divisible by 3, and so on up to all ten digits making a number divisible by 10. I figured it out using mostly guess and check, but it took a long time. Is there a quicker way? Quadrilaterals and Inscribed Circle [05/06/1999] From ten sticks of lengths 1,2,3,....,10 four are selected to form the sides of a quadrilateral... Quadrilaterals in a 3x3 Array of Dots [03/10/1999] Counting them with combinatorics, then taking away degenerate cases. Random Card Shuffling Probabilities [6/11/1996] What is the probability of at least two eights being next to each other in a random shuffling of a deck of cards. What about at least two cards (2 eights or 2 queens etc.) being next to each other? Rectangles on a Chessboard [02/09/2002] How many rectangles are there on a chessboard? Re-seating a Thousand People [6/30/1996] Can a thousand people seated around a circle in seats from 1 to 1000 be re-seated so as to preserve their order but with no person's number same as that of his chair? Reverse Permutation Doubt [12/14/2012] A student worries that a permutation represents the reverse of the seating assignment problem at hand. Doctor Peterson suggests a change in perspective to help avoid such confusion. Rock, Paper, Scissors [03/29/2001] If three people are playing Rock-Paper-Scissors, how many different combinations can be made, assuming order doesn't matter? Rubik's Cube Combinations [04/11/2001] I read that a rubics cube has 4 quintillion different possible combinations. Is this number correct? How can I calculate this value on my own? Rugby Order [08/23/2003] If there are 8 players and 5 positions, making 6720 different arrangements possible, how is that number reduced if 2 of the players can only play on the very outside right, and 1 can only play in the middle? Rugby problem [07/25/1997] What is the smallest percentage of players who are casualties of all four injuries... ? Seating Arrangements [05/16/1997] If m indistiguishable men and w indistinguishable women sit around a round table, how many possible seating arrangements are there? Seating Arrangements [02/28/1999] What is the average distance between the members of a particular couple...? Seating People in a Row [12/16/1995] Find the numbers of ways in which 4 boys and 4 girls can be seated in a row of 8 seats if they sit alternately... Sets and Subsets [03/11/2003] Consider a collection of 26 stones weighing 1, 2, 3, … , 26 grams. Prove that any subset consisting of at least 7 stones contains two separate subsets with equal total weights. Seven Elevators Stop at Six Floors [09/14/2002] A building has 7 elevators, each stopping on at most 6 floors. If you take the right elevator you can get to any floor from any other floor without changing elevators. What is the greatest number of floors the building can have? Shaking Hands - How many were at the party? [6/20/1996] Each person shook hands with everyone else. Mr. Li shook hands with 3 times as many men as women. Mrs. Li shook hands with 4 times as many men as women. Shirts and Pants [08/29/1997] Ed has 6 new shirts and 4 new pairs of pants... How many combinations of shirts and pants does he have? Social Insurance Number [7/3/1996] Make a valid Social Insurance Number that has 8 as its check digit. Squares on a Checkerboard [04/26/1998] How many squares are there on a checkerboard? Squares, Rectangles on a Chessboard [08/14/1997] How many squares are there on a chessboard? How many rectangles? Stanley Cup Finals [10/26/1999] In a Stanley Cup final, team A and team B play until one team wins four games. How many different Stanley Cup finals are possible? Stirling Numbers [05/26/1999] Can you show how to evaluate Stirling Numbers of the first and second kinds? Stirling's Approximation [05/16/2001] Is there a way to get the answer to a factorial without having to multiply out all the numbers? Page: []

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