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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?

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?

Subsets of Combinations [10/05/2003]

How many subsets of size 10 of the numbers 1, 2, 3, ..., 20 have 5 as the smallest element?

Summing n^k [11/24/1998]

Is there a general formula for summing the n^k, where k is a positive integer?

Sum of First Four Digits Equals Units Digit [10/20/2001]

How many even five-digit numbers have the property that the sum of the first four digits is the units digit?

Sum of Integers in a Set [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.

Sum of Natural Numbers [11/23/1997]

Is there a formula for counting all the possibilities for writing a natural number as the sum of natural numbers?

Tennis Match Winners [07/18/2003]

There are 7 tennis matches this weekend, of which there can only be a win or loss in each match.How many different combinations of winners can there be?

Theory of 'Runs' [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...?

The Three Canteens [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...

The Three Little Pigs Arrange Their Houses [05/13/2002]

How many different ways could the pigs arrange their houses? I come up with 27 by using 3*3*3 = 27. But when we go to list the different combinations we can only get 6.

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