Permutations and Combinations

• Choose Three People from Five [Barbara, 05/09/2001]
  What do the fractions in the numerator and denominator mean? Where does the formula come from?

• Choosing Computer Disks [Julie, 12/01/2001]
  A bin of computer disks contains a supply of disks from four different manufacturers. In how many ways can you choose 6 disks from the bin?

• Combinations of Images on a TV Screen [Kent, 12/6/1995]
  I'd like to know how many images would be possible on a hypothetical TV screen.

• Combinations of the Letters in a Name [Sara, 03/04/1998]
  A formula for the number of combinations of the letters in a name, however many times one letter appears in that name.

• Combinatorics Basics [Scamacca, 07/07/1997]
  I need to prove that n chooses n-1 = n; e.g. C(n,n-1) = n.

• Committee of Six [Samantha, 10/07/1999]
  A club has 8 male and 8 female members and is choosing a committee of 6 members, 3 male and 3 female. How many different committees can be chosen?

• Counting Answer Keys [Vickery, 05/25/2001]
  A multiple-choice test has 30 questions, each with five choices. How many answer keys are possible?

• Handshakes and Polygon Diagonals [Brendan, 09/12/2001]
  If a polygon has 42 sides, how many diagonals does it have?

• Math Symbol for C [Valovcin, 12/15/1997]
  I am puzzled by one symbol of typing math. What does the upper case letter C mean? As in (2C1) (3C1) / (47C2) = 6/1081.

• Number and Letter Combinations [Chilcot, 07/29/1997]
  How many possible combinations can a person make with the letters a-z and 1-9 starting from 1 digit and ending up with up to 8 digits?

• Number of Combinations for 7 Dice [Bertler, 2/22/1995]
  I'm looking for the derived formula solution for the following problem someone posted in a math echo: There are seven dice. Each die has six faces. How many different combinations are there of these seven dice?

• Number of Possible Telephone Numbers [Yulam, 7/15/1996]
  How many phone numbers are there given the following restrictions on certain digits of the number....?

• Odds vs. Probability [Peace, 12/06/1996]
  What's the probability that a man can draw the numbers 2 and 7 from a hat containing the numbers 1-8?

• Pascal's Triangle and Binomial Coefficients [Michelle, 09/11/2001]
  Why are Pascal's triangle and the binomial coefficients the same?

• Pascal's Triangle and the Tower of Hanoi [Flynn, 08/07/1997]
  There's a connection between Pascal's Triangle and the Tower of Hanoi game but I can't remember what it is!

• Permutations and Combinations: Deriving Formulae [Lim, 06/18/1998]
  I have been trying to figure out how to arrive at the formula for calculating conbinations.

• Possible Letter Arrangements [Stanley, 07/05/2000]
  Is there a formula I can use to quickly get the number of possible arrangements of five, six,or seven different letters?

• Powerball Lottery: Odds of Winning a Prize [Joshua, 09/02/2001]
  How are the Powerball lottery odds established?

• Round Robin Tournament Schedule [Kinley, 03/31/2000]
  Is there a systematic way to come up with a schedule for a round robin tournament for up to 32 teams, where each team plays every other team once?

• Ways To List 1, 2, ... 10 Out of Order [Brennan, 02/25/1997]
  How many ways are there to list the numbers one through ten so that no number appears in its own position (i.e. 1 is not first in the list, 2 is not second...)?

• Permutation Formula [Grech, 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.

• Permutations of Lottery Numbers [RNJM1, 8/5/1995]
  Our new lotto will start soon and the object is to pick 5 numbers, 1 through 39. No repeat or bonus numbers. How can I get a listing of all 575,000+ possible numbers?

• Winning the UK National Lottery [Dominic, 11/8/1995]
  I need to know the odds on the UK National Lottery: punters choose 6 different numbers between 1 and 49. How many permutations? How can I calculate the reduced number of permutations to reflect the fact that the order of the chosen numbers is not relevant?

• Combinations for Casting a Play [Pennell, 11/15/1995]
  Here's a list of people trying out for parts in the school play. If someone does not get a part he or she may try out for another. How many possible casts are there?

• Seating People in a Row [Lasalle, 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...

• Total of Possible Telephone Numbers [Rkbj, 1/23/1996]
  Try to determine the total possible telephone numbers in the U.S. using the following: Area codes are three digits in which the middle digit is either 0 or 1 and the first digit can be anything but 0 or 1. No three exchange digits can be the same as the preceding area code.

• Maximum Possible Combinations [Rues, 1/23/1996]
  Which is the maximum number of possibilities that I can obtain for n=digit?

• Combinations of Letters and Numbers [Hansen, 3/19/1996]
  A moped license plate has two letters and then four numbers in it. How many plates can be made without duplicating?

• Proof by Induction [Sylvestr, 4/3/1996]
  I was given a proof by my math teacher: by mathemetical induction, prove that i(nCi) = n2^n-1.

• Possible Combinations on a Push Lock [Shirt, 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.

• Multiplying Groups of Numbers [Sasson, 5/9/1996]
  Arrange the nine digits 1-9 into three groups...

• Lottery Permutations [Dellinger, 5/16/1996]
  Determine the number of unique permutations of a lottery where you pick 6 numbers out of 49, and numbers may not repeat...

• Combinatorics: Unique Groupings [Corin, 5/30/1996]
  Twenty-four friends want to play as many rounds of golf as they can... how many unique rounds of golf can they play?

• Combinations and [Bond, 6/11/1996]
  There are 6 boys and four girls. The boys do not have to stand together but they must be in order from shortest to tallest. Of these ten people, how many ways can they line up together?

• Random Card Shuffling Probabilities [Brasher, 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?

• Combinatorial Proof [Brasher, 6/13/1996]
  Please prove this combinatorial proof.

• Connect 4 - Number of Winning Arrangements [HFSchool, 6/20/1996]
  Can you help us find the formula for the number of winning lines on a 7 x 6 Connect-4 board?

• Shaking Hands - How many were at the party? [Cheng, 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.

• Handshakes at a Party [Stuart, 6/23/1996]
  At a party, everybody shakes hands with each other once. There are 45 handshakes. How many people are there at the party?

• Re-seating a Thousand People [Gellman, 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?

• Social Insurance Number [Steven, 7/3/1996]
  Make a valid Social Insurance Number that has 8 as its check digit.

• Combinatorial Proof [Turner, 7/16/1996]
  How do I prove that C(n,r)C(r,k) = C(n,k)C(n-k,r-k) where k <= r <= n?

• Unique Subset of Set of Fractions [Peters, 7/19/1996]
  How can I determine a set of fractions such that if I add any subset of those fractions, I get a result that is unique relative to the result of any other subset in this set?

• Digits that Average [Chiu, 7/19/1996]
  How many integers from 100 to 999 inclusive have one digit that is the average of the other two?

• Partitions of Set {1,2,3, ...n} [Scott, 7/25/1996]
  Is there a generic way or some formula to count the partitions of the set {1,2,3, ...n}?

• Full House in Draw Poker [Ludwig, 7/28/1996]
  A player is dealt 5 cards - a pair of aces and three other unpaired cards. What is the probability of winding up with a full house...

• Lottery Combinations [Felix, 8/20/1996]
  How do you compute the number of the lottery combinations for 6 of 49 numbers?

• Theory of 'Runs' [Brasher, 8/20/1996]
  If all possible orders of 20 people are considered, what is the average value of the number of places in the row...?

• Combinations of Three Words [Foster, 8/28/1996]
  I have three columns of 20 words each...

• Choose 11 Football Players in 12,367 Ways [Cowley, 9/16/1996]
  A gym coach must select 11 seniors to play on a football team. If he can make his selection in 12,376 ways, how many seniors are eligible to play?

• Permutations and Combinations [Jain, 10/22/1996]
  How many different six-digit numbers can be formed using three 5's, two 4's, and one 6?

• Pieces on a Chess Board [Ho, 10/27/1996]
  Prove that with 9 seperate 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.

• Combinations of Cubes [Whistler, 11/07/1996]
  How many different cuboids can be made from one million connectable cubes, using all the cubes?

• Combinations of Married Couples [Chu, 11/08/1996]
  What is the probability that 12 people can be grouped into 6 pairs where each pair is a married couple?

• Enumeration in Combinatorial Analysis [Mauer, 11/28/1996]
  How many 3 digit numbers have the digital sum of nine?

• Combinations of Pegs [Parnell, 02/04/1997]
  Given a square peg board with sixteen pegs, how many triangles can you form by connecting three pegs?

• 20 Golf Players [Mullen, 04/11/1997]
  How do you set up four teams such that the variance in the average team handicap is no more than one or two?

• Divisibility by Four [Unruh, 04/19/1997]
  How many 6-digit numbers are divisible by 4, if we allow no repeated digits?

• Permutations of Vertices [Steen, 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?

• Beads on a Bracelet [Buhler, 05/04/1997]
  How many different arrangements of 3 red and 3 blue beads on a bracelet are there?

• Handshake Problem [Bushnell, 05/07/1997]
  If you have n people in a group and each shakes everyone else's hand only once, how many handshakes take place?

• Poker Combinations [Fried, 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?

• Seating Arrangements [Lerche, 05/16/1997]
  If m indistiguishable men and w indistinguishable women sit around a round table, how many possibile seating arrangements are there?

• MathCounts Problem [Li, 07/18/1997]
  Two boys and four girls are officers of the Math Club. With the faculty sponsor in the middle and the two boys not next to one another, how many different seating arrangements are possible?

• Rugby problem [Holman, 07/25/1997]
  What is the smallest percentage of players who are casualties of all four injuries... ?

• Math Poster: Handshakes [Vitale, 08/11/1997]
  If there are 15 people in the room and each person shakes hands with every other person, how many handshakes will there be?

• Combinatorics [Chaudhuri, 08/19/1997]
  sum((-1)^k*((4n choose 2k)/(2n choose k)) for k = 1 to 2n

• Shirts and Pants [Spark, 08/29/1997]
  Ed has 6 new shirts and 4 new pairs of pants... How many combinations of shirts and pants does he have?

• Combinations of Prisoners [Lawrence, 09/02/1997]
  Nine prisoners are taken for their daily exercise handcuffed together in threes. How would the warden arrange the men each day so that no two men are handcuffed together more than once over a six day period?

• Why is 0! 1? [Kimberly, 09/14/1997]
  Why is zerofactoro 1?

• Brick Patterns [Jessi, 09/17/1997]
  How many different ways can 1x2 ft. bricks be arranged to make a 20' long path?

• The Three Canteens [Joe, 09/18/1997]
  You have only one 14-cup canteen full of water, and two empty canteens that will hold nine and five cups respectively...

• Checkerboard Squares [Gibson, 09/29/1997]
  Suppose you have a square checkerboard not 8x8 - what is the easiest way to find how many squares there are in it?

• Permutations [Coelho, 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.

• Poker, Probability, Combinatorics [Hundal, 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?

• Divisors [Ryan, 11/07/1997]
  What kinds of numbers have exactly 3 divisors? Do bigger numbers necessarily have more divisors? Is there a way to figure out how many divisors 1,000,000 has without actually listing and counting them?...

• What's the Probability That Two Will Agree? [Hawkins, 11/10/1997]
  Of the nine members of the board of trustees of a college, five agree with the president. The president selects three trustees at random and asks their opinions...

• Sum of Natural Numbers [Lerche, 11/23/1997]
  Is there a formula for counting all the possibilities for writing a natural number as the sum of natural numbers?

• Palindromic Numbers [Akyildiz, 01/04/1998]
  How many palindromic numbers exist that are less than 100,000?

• What Number Falls on the 124th Position? [Kate, 01/06/1998]
  There are 720 (or 6!) permutations of the digits 1,2,3,4,5,6. If these permutations are arranged from smallest to largest numerical value... (and other questions).

• Marathon Prizes [Abahna, 01/10/1998]
  Ten runners are competing in a marathon. In how many ways can the first and second prizes be awarded?

• Combinatorics: Ramsey Theory [Busardo, 01/12/1998]
  Could you help me with a detailed explanation of the theory and a concrete example?

• How Many Different Boxes of Donuts Can Be Made? [Ferreira, 03/01/1998]
  Using the "stars and bars" method to count how many different boxes of a dozen donuts can be made from five different types of donuts.

• Permutations and Combinations [Demers, 03/11/1998]
  What is the difference between permutations and combinations?

• Possible Combinations from Cards Drawn at the Same Time? [gladys, 04/02/1998]
  If two of four numbered cards are drawn at the same time, how many possible combinations could be made?

• Drawing Marbles [West, 05/05/1998]
  A jar contains 19 different marbles. I draw 13 marbles, one at a time, but replace the marble back before taking the next....

• Combinations of Poker Hands [Bass, 05/15/1998]
  Counting three of a kind, two pair, and one pair poker hands.

• Pascal's Triangle and Combinations [Walker, 06/05/1998]
  Can you show me why (n C r) + (n C r+1) = n+1 C r+1?

• Arranging Marbles in Boxes [Chris, 09/16/1998]
  Six marbles are placed in one of three different boxes. What is the probability that each box contains two marbles?

• Product of Disjoint Cycles [Pelayo, 10/16/1998]
  How to express (1 2 3 5 7)(2 4 7 6) as the product of disjoint cycles.

• Counting Digits [Erle, 10/23/1998]
  Using the Fundamental Principle of Counting, how many six-digit numbers can you make with two zeros, two twos, and two fours?

• Combinations of Letters [Anthony, 10/20/1998]
  What is the general formula for counting the number of different ways to combine the letters of one word? For example, Emma has 12 ways.

• Blocks of Seats [Mike, 11/08/1998]
  In a row of 20 seats, in how many ways can 3 blocks of consecutive seats with 5 seats in each block be selected? And more such problems...

• Arranging Letters [Mike, 11/08/1998]
  In how many ways can the letters in MISSISSIPPI be arranged? Suppose the 2 P's must be separated?... and other such problems.

• Partitioning an Integer [Huckin, 11/14/1998]
  How many different ways are there of making a number by adding different combinations of three numbers?

• Summing n^k [Kijjaz, 11/24/1998]
  Is there a general formula for summing the n^k, where k is a positive integer?

• Permutation, Combination, and Repetition [Santos, 11/25/1998]
  What is a permutation and what is a combination with repetition and no repetition?

• Word Derangements and Arrangements [Julie, 11/30/1998]
  I need to find a formula to count the number of derangements of a word. First you start with the number of ways to arrange the letters...

• Trinomial Expansion [Achille, 12/08/1998]
  I need to expand (1+x+x^2)^n to four terms. Is there a formula?

• Coin Tosses, Dealing Cards... [Greg, 12/08/1998]
  Several questions on discrete math - probability and combination; deducing recurrence relations.

• Seating Arrangements [Nicolas, 02/28/1999]
  What is the average distance between the members of a particular couple...?

• Change for a Dollar [Jay, 03/07/1999]
  How many different ways are there to make a change for a dollar?

• Quadrilaterals in a 3x3 Array of Dots [David, 03/10/1999]
  Counting them with combinatorics, then taking away degenerate cases.

• Connecting the Dots [Tim, 03/14/1999]
  If you have a few dots on a page, how many lines does it take to connect them all to each other?

• Letters and Envelopes - the Inclusion-Exclusion Principle [Jarrad, 03/27/1999]
  What is the probability that no rabbit will escape into its own hole?

• Binomial Probability [Pacyon, 04/14/1999]
  A multiple-choice exam probability question.

• Paths on a Checkerboard [mike, 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?

• Balls in Boxes [Bassa, 05/05/1999]
  You have 2*t + 1 balls to put into 3 boxes, but the sum of the balls in 2 of the boxes should be more then the balls in the other box...

• Quadrilaterals and Inscribed Circle [Jayant, 05/06/1999]
  From ten sticks of lengths 1,2,3,....,10 four are selected to form the sides of a quadrilateral...

• Vertices in a Prism [Bergner, 05/12/1999]
  What is the formula for finding the number of vertices in a prism?

• Choosing Socks [Alume, 05/20/1999]
  What are the steps to evaluate C(n,1); C(n,2) ?

• Twelve Beads on a String [McCasker, 05/24/1999]
  How many different arrangements are there when there are 12 beads of 4 different colours on a string?

• Paths to Triangle Points [Goldstein, 05/26/1999]
  How can I find the number of paths to a point using Pascal's triangle?

• Stirling Numbers [John, 05/26/1999]
  Can you show how to evaluate Stirling Numbers of the first and second kinds?

• Counting Triangles [Michael, 05/27/1999]
  In a large triangle with 36 small ones inside, how many triangles are there in all?

• Tie Combinations [Feller, 05/31/1999]
  If four people have three identical ties, what is the total number of different combinations the four can wear, and what is the probability that all four people will wear the same tie?

• Two Combination Problems [Devlin, 06/04/1999]
  How many ways are there to distribute 40 identical items among 4 people? How many different subcollections of n objects are there in a collection of 2n objects, n identical and the other n distinct?

• Combinations with Duplicate Objects [Black, 06/22/1999]
  How many different combinations are there when choosing 3 letters from the group {ABBCCC}?

• Permutations in a Necklace [Romer, 07/04/1999]
  What is the formula for the number of permutations in a necklace with the combination of AABB?

• Handshake Problem Variant [Bixler, 07/08/1999]
  Five couples go to a party and start shaking hands. One of the men shouts, "Stop! How many hands did you shake?" Every person gives a different answer...

• Arrangement [Adge, 07/12/1999]
  How can I reduce the number of arrangements of the word ARRANGEMENT by using the probability of an occurrence?

• Permutation and Combination Equality [Carlos, 07/18/1999]
  Prove that (nC0)^2 + (nC1)^2 + (nC2)^2 + ... + (nCn)^2 = (2nCn), where nCi = n!/((n-i)!*i!).

• Proof of the Addition Principle by Induction [Thomas, 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?

• Co-efficient of an Algebraic Term [Elizabeth, 07/18/1999]
  Simplify [z^40] (1 + z + z^2 + ... + z^9)^100.

• How Many Barbies [Huron, 08/11/1999]
  How many different Barbies can I make if I can choose from 4 skin colors, 3 eye colors, 4 hairstyles, and 6 hair colors?

• Maximizing the Product of Partition Elements [Jacob, 08/20/1999]
  How can you prove that the product of the elements in a partition is at a maximum when the elements are equal?

• Four-Letter Combinations [Singh, 09/06/99]
  How can I find the number of four-letter combinations of the letters in the words COLLEGE and PRINCIPAL?

• Permutations of Beads on a Necklace [Romer, 09/07/99]
  What is the formula for the number of permutations in a necklace with the combination of AAABBB?

• Probability and Permutations [Sunda, 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?

• Arrangements of Letters [Kevin, 09/14/1999]
  In how many ways can the letters in UNUNSUAL be arranged? For those arrangements, how many have all 3 U's together?

• Combinations Totaling 100 [Alan, 09/27/1999]
  In how many ways can I achieve a sum of 100 adding together only 6 integers taken from the set of integers from 1 to 44?

• Path Possibilities [Roger, 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?

• Stanley Cup Finals [Cote-Julien, 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?

• Combinations of X's and Y's [Dodgson, 10/27/1999]
  X's and Y's are written in a row (e.g. XX...XXYY...Y). How many different arrangements of the letters can there be?

• Cent mille milliards de poemes [Sarah, 02/12/2000]
  Question submitted via WWW: Hello. I have this problem and I've been doing research to find the answer but I've had no luck. How did Queneau fit 10^14 poems into ten pages?

• Choosing 3 of 6 Colors [Marcy, 03/03/2000]
  Patrick has a box of crayons with red, blue, yellow, orange, green and purple. How many different ways can Patrick select 3 colors?

• Counting Intersections of Diagonals in Polygons [Dingena, 03/08/2000]
  Can you help me find an equation for the maximum number of intersections of the diagonals in a polygon?

• Hockey League Tournament Schedule [Gibbons, 03/09/2000]
  Can you help me construct a schedule for a round robin style junior hockey tournament with two divisions (one of 6 teams, one of 10), using only four fields?

• Identical/Non-identical Groups; Derangements [Kala, 03/10/2000]
  Formulas for dividing things into identical or different, non- identical groups; derangements.

• Collecting a Complete Set [Butler, 03/15/2000]
  I have a very large box filled with 8 different frying pan handles in equal proportions. What is the probability that I will have to remove 32 handles before I get a complete set?

• Generating All Permutations of a Set [Harris, 05/11/2000]
  How can I write out all the permutations of a set of objects in a non- recursive way?

• Optimal Seating Arrangements [Schwartz, 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?

• Permutations of Indistinguishable Objects [Candice, 09/21/2000]
  How many ways can eight dogwoods be planted?

• Induction With Binomial Coefficients [Lee, 10/16/2000]
  Prove that the sum from i = 1 to n of (i+k-1 choose k) equals (n+k choose k+1).

• Dinner Triplets [Reagan, 10/23/2000]
  A woman has 15 friends. For 35 days she wants to have dinner with 3 friends a day, arranging it so that each pair of friends will come only once. Is this possible?

• How Many Distinct Patterns? [Ralph, 01/15/2001]
  Given a large equilateral triangle divided into four smaller equilateral triangles, if two edges are painted white and the rest are painted black, how many distinct patterns are possible?

• Lab Partner Pairings [Maynard, 01/21/2001]
  A teacher would like to find all the possible lab pairings of his 22 students. Is there an algorithm for this?

• How Many Handshakes? [Kathy, 01/29/2001]
  There are 40 people in a room. They shake each other's hands once and only once. How many handshakes are there altogether?

• Gray Code [Hooman, 02/16/2001]
  What is Gray code? How does it relate lists of n-tuples of 0's and 1's to traversing the corners of an n-dimensional cube?

• Examples of the Fundamental Counting Principle [Diana, 02/17/2001]
  There are three ways to go from Town A to Town B, and four ways to go from Town B to Town C. How many different ways are there to go from Town A to Town C, passing through Town B?

• Ways to Pick a Six-Card Hand [Shaw, 02/26/2001]
  How many ways are there of picking six cards from a deck, where suit and order of selection are ignored?

• Numbering the Faces of Dice [Cotterill, 02/27/2001]
  How many ways are there to make dice out of the Platonic solids (i.e. 4, 6, 8, 12, and 20 sides)? How many of those ways have opposite face sums equal? What would the opposing face sums be for each type?

• Checkout Registers and Customers [Kirkman, 02/27/2001]
  Given two checkout registers and twenty customers, find a formula to calculate how many different ways they can be arranged (order matters).

• NCAA Tournament Possibilities [Jill, 03/14/2001]
  In an NCAA Tournament office pool where you fill out the brackets by selecting the team you think will win each game, how many possible combinations are there?

• Checkerboard Combinations [Ryan, 03/21/2001]
  What is the possible number of combinations on a checkerboard with 12 (or fewer) red pieces and 12 (or fewer) black pieces?

• Painted Cube Faces [Paul, 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?

• Rock, Paper, Scissors [Baum, 03/29/2001]
  If three people are playing Rock-Paper-Scissors, how many different combinations can be made, assuming order doesn't matter?

• Probability of Never Having a Losing Record [Bulut, 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?

• Rubik's Cube Combinations [Glapa, 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?

• Coefficients in a Trinomial Expansion [Birkenfeld, 04/24/2001]
  In the expansion of (a+b+c)^6, what is the coefficient of a^2b^2c^2?

• Placing Balls in Urns [Jonathan, 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).

• Stirling's Approximation [Faith, 05/16/2001]
  Is there a way to get the answer to a factorial without having to multiply out all the numbers?

• Counting Rectangles [Calabrese, 05/23/2001]
  How can I find the number of different rectangles in a square grid containing "c" columns and "r" rows?

• Importance of Surface Area [Francine, 05/26/2001]
  Why is surface area so important? What kinds of things depend on surface area?

• Making Change for a Dollar [Kathy, 05/29/2001]
  What is the smallest number of coins you CANNOT make change with?

• Calculating Permutations [Andrew, 05/29/2001]
  Please explain the formula for calculating the total possible number of ways to choose r objects out of a total n objects.

• Choosing Three Numbers from 1-10 [Craig, 06/08/2001]
  How many ways can three numbers be chosen from the numbers 1,2,3,4,5,6,7,8,9,10 so that no two of the three numbers are consecutive?

• Sum of Integers in a Set [Martin, 06/19/2001]
  Consider the set of all four-digit integers, each of which is formed using the digits 1,2,3,5, or 7 at most once. Find the sum of the integers in this set.

• Painted Cube [Ste, 06/21/2001]
  I've done diagrams and tables, but just can't see or find a formula for the problem.

• Two People, Three Objects [Nupur, 07/14/2001]
  Find the number of ways in which 2n objects of type A, 2n objects of type B, and 2n objects of type C can be divided between two persons, giving 3n objects to each.

• Proof of Ordered Partioning of Integers [Dina, 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.

• Occupancy Problem [Mauricio, 08/06/2001]
  Given n bins and m (indistinguishable) balls, how many arrangements are possible such that no bin has greater than r balls?

• Distributing Objects [Ali, 08/12/2001]
  Is there a formula for distributing n indistinguishable objects into k indistinguishable urns?

• Bell Numbers [Ben, 08/29/2001]
  I am looking for the formula for the number of different groups we can split a group of n different items into - order does not matter.

• Pascal's Triangle: Words instead of Numbers [Ryan, 08/28/2001]
  How many times can you read the word "triangles" in the figure?

• Number/Color Cube [Bill, 09/13/2001]
  You want to make a number cube by putting the numbers 1,2,3,4,5,6 on the face. 1/5, 3/6, and 2/4 must be on opposite faces. Each face is a different color. How many ways can you make the cube?

• No Three Red Beads Together [Jenny, 09/16/2001]
  Given 10 beads on a necklace, 6 white and 4 red, how many ways can the beads be arranged so that no three red beads are together?

• Martha and Four Friends [Vidya, 09/17/2001]
  Martha and four friends go to a movie. How many different ways can they sit together with Martha always between two friends?

• Permutations of 1234567890 [Leeanna, 09/23/2001]
  I'm trying to find a pattern in combinations and how many combinations there are in 1234567890.

• Baseball Batting Orders [Erica, 10/14/2001]
  How many possible batting orders are there for a team of nine players?

• Four Colors, Eighths of a Circle [Steve, 10/15/2001]
  Divide a circle in eighths. Use 4 colors to color the segments. Colors may be repeated as long as you use all 4 colors at least once. What are the total combinations possible?

• Minimum Set of Weights Puzzle [Jhonen, 10/18/2001]
  What is the minimum number of weights needed for a scale that can weigh objects from 1-100 pounds, inclusive, at one-pound increments?

• Sum of First Four Digits Equals Units Digit [James, 10/20/2001]
  How many even five-digit numbers have the property that the sum of the first four digits is the units digit?

• Cuisenaire Rod Combinations [Betty, 11/13/2001]
  We are to record 512 different combinations of each level (or color) of rod, with the orange rod (the largest) being the last. Is there a formula exist to find these combinations without literally manipulating the rods?

• Color Arrangements on the Face of a Cube [Jhonen, 11/23/2001]
  Given three colors in which to paint the face of a cube, red, blue, or yellow. How many different color patterns are there if each face of the cube must be painted red, blue, or yellow?

• Traveling Through a Square [Stephen, 11/25/2001]
  How do I get from the bottom left-hand corner of a 64-block square to the top right-hand corner, only going through each square once?

• Laying Paving Stones [Brian, 11/28/2001]
  Finding a relation for a sequence that relates to the number of ways paving stones can be laid to make a 3-foot-wide path using 3-foot by 1-foot stones.

• TURKEYS puzzle [Donna, 11/28/2001]
  Count the number of ways you can trace the word TURKEYS in the triangular array. You may only move to one of the two letters directly below the letter you are on.

• Partitioning Elements [Rajshree, 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)...

• English Exam Probability [Hunter, 12/17/2001]
  Given a list of 25 potential exam questions, with 10 of them on the actual exam and of which students write essays on 2, how many questions should be prepared to be 95% confident you won't write on a question you haven't prepared?

• How Many Games in the Tournament? [Crumpy, 01/15/2002]
  There are eight teams in a single-elimination tournament. Each team gets to play until it loses. How many games will be played in the tournament?

• Colour Combinations on a Cube [Daniel, 02/08/2002]
  If each side of a cube is painted red or yellow or blue, how many distinct colour patterns are possible?

• Bell Number [William, 02/24/2002]
  I am trying to figure out how many different groups 50 people can be partitioned into.

• Generating Eight-Character Passwords [Lohkee, 03/08/2002]
  Given some restrictions, calculate the number of possible 8-character passwords.

• Arrangements of 0's, 1's, and 2's [Sarah, 03/14/2002]
  How many arrangements of six 0's, five 1's, and four 2's are there in which i) the first 0 precedes the first 1? ii) the first 0 precedes the first 1, precedes the first 2?

• Buying Doughnuts [Wendy, 03/22/2002]
  Janine wants to buy three doughnuts, and there are five varieties to choose from. She wants each doughnut to be a different variety. How many combinations are there?