See also the
Dr. Math FAQ:
Browse High School Permutations/Combinations
Stars indicate particularly interesting answers or
good places to begin browsing.
- 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.
- Tie Combinations [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?
- Tiling a Mutilated Chessboard With Dominoes [08/29/2003]
Suppose we take an ordinary chess board and randomly remove a
black square and a white square. Is it always possible to cover what
remains with 2x1 dominoes? If yes, how? If no, why not?
- Total of Possible Telephone Numbers [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.
- Traveling Through a Square [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?
- Triangle Vertices But Not Sides [02/22/2003]
If P is a regular n-gon, what is the number of triangles whose
vertices are the vertices of P but whose sides are NOT the sides of P?
- Trinomial Expansion [12/08/1998]
I need to expand (1+x+x^2)^n to four terms. Is there a formula?
- Trinomial Expansion [02/23/2003]
How can I find the number of terms of (a + b + c)^140?
- Trying to Unlock a Cell Phone [11/16/2003]
I'm trying to unlock a cell phone that has a four-digit number as the
code. How many possible codes are there for me to try?
- TURKEYS puzzle [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.
- Twelve Beads on a String [05/24/1999]
How many different arrangements are there when there are 12 beads of 4
different colours on a string?
- Two Combination Problems [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?
- Two People, Three Objects [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.
- Understanding Probability [05/13/2002]
If I were looking for a probability of colored marbles in three
packages, how would I do that?
- Unique Subset of Set of Fractions [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?
- Version of Lights-Out Puzzle Using Buttons [09/02/2004]
A variation on the classic Lights-Out Puzzle in which pushing any
button in a 6x6 grid changes the state of that button and all others
in the same row or column. The goal is to maximize the number of
buttons in a given state.
- Vertices in a Prism [05/12/1999]
What is the formula for finding the number of vertices in a prism?
- Ways to Pick a Six-Card Hand [02/26/2001]
How many ways are there of picking six cards from a deck, where suit and
order of selection are ignored?
- What Is N Factorial Used For? [08/29/2001]
I have heard that it is used in probability, but what for, and how?
- What Number Falls on the 124th Position? [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).
- What's the Probability That Two Will Agree? [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...
- Why is 0! 1? [09/14/1997]
Why is zero factorial 1?
- Why Is the Number of Combinations Equal? [06/13/2007]
Can you explain why nCr = nC(n-r), such as 7C2 = 7C5 = 21?
- Winning the Lottery [07/22/2009]
Choose 5 numbers from 1-40 and match 3 randomly picked numbers from
1-40 to win. How many different combinations of 5 numbers are there in
40 and how many different guesses of 5 would it take to cover every
possible outcome of three numbers?
- Winning the UK National Lottery [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?
- Word Derangements and Arrangements [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...