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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. Occupancy Problem [08/06/2001] Given n bins and m (indistinguishable) balls, how many arrangements are possible such that no bin has greater than r balls? On Spins and Surprises [10/15/2011] Nine consecutive spins of a roulette wheel surprise a gambler unfamiliar with how to determine the likelihood of independent events. Doctor Vogler obliges with the requested combinations and probabilities, but only after questioning our arbitrary preference for some events over others that may have the same -- or even longer -- odds. Onto Functions and Stirling Numbers [09/22/2002] How would I show for m greater than or equal to 3 that s(m, m-2) = (1/ 24)m(m-1)(m-2)(3m-1), where s(m,n) are Stirling numbers of the first kind? Optimal Seating Arrangements [07/20/2000] N people are invited to a party and asked to RSVP with the names of up to k people they would like to sit with. Is there a formula that will yield the "best" arrangement of people? Ordering Combinations, Then Picking the nth Item [10/22/2010] How do you pick the nth combination without listing all the ones before it? Doctor Vogler restores some order to an ambiguous objective, then outlines an algorithm for sorting combinatorics. P(2n,3) = 2P(n,4) [10/08/2002] Solve: P(2n,3) = 2P(n,4) Painted Cube [06/21/2001] I've done diagrams and tables, but can't find a formula for the problem. Painted Cube Faces [03/22/2001] Each face of a cube is to be painted either red or blue. How many different possible combinations are there, from all blue to all red faces and every combination in between? Painting Cube Faces Green [08/19/2003] How many cubes could be painted so that each cube is different from the other? Palindromic Numbers [01/04/1998] How many palindromic numbers exist that are less than 100,000? Partitioning an Integer [11/14/1998] How many different ways are there of making a number by adding different combinations of three numbers? Partitioning Elements [12/08/2001] If part(n,k) is the number of ways to partition a set of n elements into k subsets, what is Part(5,2)? Prove Part(n+1,k) = Part(n,k-1)+k* Part(n,k)... Partitions of Set {1,2,3, ...n} [7/25/1996] Is there a generic way or some formula to count the partitions of the set {1,2,3, ...n}? Pascal's Triangle and Combinations [06/05/1998] Can you show me why (n C r) + (n C r+1) = n+1 C r+1? Pascal's Triangle: Words instead of Numbers [08/28/2001] How many times can you read the word "triangles" in the figure? Path Possibilities [10/24/1999] How many combinations of moves are there to get from point A to point B in a modified checkerboard figure, assuming that you can only move right and up? Paths on a Checkerboard [04/18/1999] On an 8x8 checkerboard, how many paths are there from point A to point B following only the lines going downward and to the right? Paths to Triangle Points [05/26/1999] How can I find the number of paths to a point using Pascal's triangle? Perfect Shuffle [09/09/2002] If you were to shuffle a deck of cards perfectly, how many repetitions would it take to return to the original order? Permutation and Combination Equality [07/18/1999] Prove that (nC0)^2 + (nC1)^2 + (nC2)^2 + ... + (nCn)^2 = (2nCn), where nCi = n!/((n-i)!*i!). Permutation, Combination, and Repetition [11/25/1998] What is a permutation and what is a combination with repetition and no repetition? Permutation Formula [4/23/1995] I can't for the life of me remember the formula for calculating the total possible number of ways to choose 'r' objects out of a total 'n' objects. Permutation Groups Generated by 3-Cycles [05/14/2003] Show A_n contains every 3-cycle if n >= 3; show A_n is generated by 3- cycles for n >= 3; let r and s be fixed elements of {1, 2,..., n} for n >= 3 and show that A_n is generated by the n 'special' 3-cycles of the form (r, s, i) for 1 <= i <= n. Permutations [10/11/1997] A number of X's and a number of Y's are written in a row such as XX.....XXYY.....Y - Investigate the number of different arrangements of the letters. Permutations and Arithmetic Means [01/24/2003] Demonstrate that in the (n!) permutation of the first n integers in a table of dimension n! rows and n columns, the arithmetic mean of the squares of each column terms is equal to (n+1)(4n+2)/12, and the arithmetic mean of the crossproduct between any two columns is equal to (n+1)(3n+2)/12. Permutations and Combinations [10/22/1996] How many different six-digit numbers can be formed using three 5's, two 4's, and one 6? Permutations and Combinations [03/11/1998] What is the difference between permutations and combinations? Permutations, Combinations, Arrangements, and Strings [10/22/2007] When a coin is tossed four times, is an outcome of HTTH considered a permutation, a combination, or something else entirely? Permutations in a Necklace [07/04/1999] What is the formula for the number of permutations in a necklace with the combination of AABB? Permutations of 1234567890 [09/23/2001] I'm trying to find a pattern in combinations and how many combinations there are in 1234567890. Permutations of Beads on a Necklace [09/07/99] What is the formula for the number of permutations in a necklace with the combination of AAABBB? Permutations of Indistinguishable Objects [09/21/2000] How many ways can eight dogwoods be planted? Permutations of Vertices [04/22/1997] If the vertices of a polygon are labeled with letters, in how many different ways can a quadrilateral or a pentagon be named? Permutations or Combinations? [01/09/2003] When given a permutation or combination problem, is there anything SPECIFIC to look for to know whether it is one or the other? Phone Numbers With Duplicate Digits [03/25/2003] Someone is trying to remember a phone number but cannot remember the whole thing. He remembers 279-XXXX and that the last 4 numbers must contain a 2 and a 7 and a 9. He only has 2, 7, or 9 as digits. How many possible completions are there? Pieces on a Chess Board [10/27/1996] Prove that with 9 separate playing pieces, you cannot place the pieces on an 8 by 8 chess board such that the distance between any 2 pieces is always different. Placing Balls in Urns [05/15/2001] Prove that the number of different ways we can place b indistinguishable balls in u distinguishable urns is C(b+u-1,b) = C(b+ u-1,u-1). Poker Combinations [05/08/1997] In a standard deck of cards, how many different ways are there to get a straight, a flush, a straight flush, one pair, or two pairs? Poker, Probability, Combinatorics [11/04/1997] If we deal n hands consisting of 2 cards each, what is the probability that there will be no pairs amoung the hands? Polya's Counting Theory [09/12/2003] In how many different ways could a 6-sided die be numbered? Page: []

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