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Browse High School Permutations/Combinations
Stars indicate particularly interesting answers or
good places to begin browsing.
 How Many Handshakes? [08/18/2005]

Thirty people at a party shook hands with each other. How many
handshakes were there altogether? Draw a diagram and see if you can
establish a pattern by collecting data in a table for one, two, three,
four and five people shaking hands.
 How Many Possible Pizza Topping Combinations? [12/21/2004]

Dominos offers 13 possible pizza toppings, and each topping can be
ordered in single, double, or triple layers. How many different
possible pizzas can be ordered?
 How Many Regions are Generated? [05/29/2003]

Draw a circle and pick n points on it. Now join every point to every
other point and suppose that the points have been picked so that no
three chords go through one point (i.e. every intersection is the
intersection of exactly two chords).
 Identical/Nonidentical Groups; Derangements [03/10/2000]

Formulas for dividing things into identical or different, non identical
groups; derangements.
 Induction With Binomial Coefficients [10/16/2000]

Prove that the sum from i = 1 to n of (i+k1 choose k) equals (n+k choose
k+1).
 Integer Divisors [04/11/1997]

N has six distinct integer divisors including 1 and N. The product of
five of these is 648; find another divisor of N.
 Introduction to Permutations by Standing in Line [11/18/2004]

Six persons will stand in line. In how many different ways can they
stand in line?
 Inverse, Product of Permutations [04/27/2002]

I don't understand how to calculate the inverse or the product of
permutations.
 Josephus Problem [04/18/2003]

Every other person at a table is eliminated until there is only one
person left. Who is the survivor?
 Knights of the Round Table [07/01/1998]

If x knights are sitting at a round table, and every other one is
removed, who is the last one left sitting at the table?
 Lab Partner Pairings [01/21/2001]

A teacher would like to find all the possible lab pairings of his 22
students. Is there an algorithm for this?
 Languages, Grammars [04/20/2003]

Let L be the set of words w over the alphabet {a,b} that have an odd
number of a's and a multiple of three b's. Find a regular grammar
that generates L.
 Laying a Brick Walkway [04/22/2002]

How many different ways can I build a walkway 2' by 20' of bricks 1'
by 2'? The bricks can lie vertically and horizontally, but in no other
direction.
 Laying Paving Stones [11/28/2001]

Finding a relation for a sequence that relates to the number of ways
paving stones can be laid to make a 3footwide path using 3foot by 1
foot stones.
 Letters and Envelopes  the InclusionExclusion Principle [03/27/1999]

What is the probability that no rabbit will escape into its own hole?
 License Plate Combinations [11/26/2002]

How many different combinations of license plates are there for our
state, given that we have 6 spaces that can be (09) numbers and (az)
letters?
 Lottery Combinations [8/20/1996]

How do you compute the number of the lottery combinations for 6 of 49
numbers?
 Lottery Permutations [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...
 Making $5 Using 50 Coins [12/02/2005]

How many ways can you make $5 with 50 coins and without using dimes?
 Making Change for a Dollar [05/29/2001]

What is the smallest number of coins you CANNOT make change with?
 ManySided Dice, Lots of Sums  and Even More Combinations [01/13/2011]

Gamers want to compare the probabilities of the possible dice throws used in
their roleplaying games. Invoking the combinatorics of balls and urns, Doctor Vogler
confirms the complexity of enumerating the possibilities for nontrivial numbers of
rolls and dice.
 Marathon Prizes [01/10/1998]

Ten runners are competing in a marathon. In how many ways can the first
and second prizes be awarded?
 Martha and Four Friends [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?
 MathCounts Problem [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?
 Math Poster: Handshakes [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?
 Maximizing the Product of Partition Elements [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?
 Maximum Occupancy [12/23/2013]

How many ways can six people share four rooms that each sleeps up to four? After
carefully interpreting the question, Doctors Ian and Peterson help a combinatorics
student think through the possibilities.
 Maximum Possible Combinations [1/23/1996]

Which is the maximum number of possibilities that I can obtain for
n=digit?
 Minimal Weighings of Ten Coins to Identify the Two Counterfeits [04/18/2010]

At least how many balance scale weighings of ten coins do you need to determine the
two fakes? By applying combinatorics and keeping track of lower bounds, Doctor
Jacques provides a methodical approach.
 Minimum Set of Weights Puzzle [10/18/2001]

What is the minimum number of weights needed for a scale that can weigh
objects from 1100 pounds, inclusive, at onepound increments?
 Monkeys Typing Shakespeare: Infinity Theory [08/05/1998]

Would an infinite number of monkeys typing at random eventually produce
the entire works of Shakespeare?
 Multinomial Theorem and Coefficients of Polynomial Expansions [06/24/2007]

Could you explain how to use the multinomial theorem to find the
coefficients of various terms in the expansion of (x+y+z+a+b)^4?
 Multiplying Groups of Numbers [5/9/1996]

Arrange the nine digits 19 into three groups...
 NCAA Tournament Possibilities [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?
 No Three Red Beads Together [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?
 No Two Consecutive Terms Divisible by 2 or 3 [10/13/2002]

In how many ways can numbers in the set {1,2,3,4,5,6} be ordered so
that no two consecutive terms have a sum that is divisible by 2 or 3?
 Number/Color Cube [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?
 Numbering the Faces of Dice [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?
 Number of Numbers of Five Digits Only [05/29/2014]

A teen struggles to enumerate the numbers greater than 250 composed of only the
digits 1, 2, 3, 4, and 5. By reframing the question, Doctor Ian reduces a large brute
force search into a small one and a few permutations.
 The Number of Possible Sudoku Puzzles [03/04/2006]

I calculated that there are 16,930,529,280 different possible Sudoku
puzzles, but an article I saw said that there are many more. How many
are there, and how would you determine it?
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