See also the
Dr. Math FAQ:
Browse High School Permutations/Combinations
Stars indicate particularly interesting answers or
good places to begin browsing.
- 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?
- 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
- 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
- Rugby problem [07/25/1997]
What is the smallest percentage of players who are casualties of all four
- 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
- 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
- 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
- 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
- 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...