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Browse High School Permutations/Combinations
Stars indicate particularly interesting answers or
good places to begin browsing.
 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 11 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^n1.
 Proof of Ordered Partioning of Integers [07/31/2001]

I have found that there are 2^(n1) 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 09
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?
 Reseating a Thousand People [6/30/1996]

Can a thousand people seated around a circle in seats from 1 to 1000 be
reseated 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 RockPaperScissors, 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 fivedigit 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 fourdigit 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?
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