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Browse High School Permutations/Combinations

Stars indicate particularly interesting answers or good places to begin browsing.



Induction With Binomial Coefficients [10/16/2000]
Prove that the sum from i = 1 to n of (i+k-1 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 3-foot-wide path using 3-foot by 1- foot stones.

Letters and Envelopes - the Inclusion-Exclusion 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 (0-9) numbers and (a-z) 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?

Many-Sided 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 role-playing games. Invoking the combinatorics of balls and urns, Doctor Vogler confirms the complexity of enumerating the possibilities for non-trivial 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 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 1-100 pounds, inclusive, at one-pound 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 1-9 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?

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?

Number of Terms in a Polynomial Expansion [12/01/2005]
How would you calculate the total number of terms in the simplified expansion of (a+b+c+d)^10? Is there a general formula for this?

Occupancy Problem [08/06/2001]
Given n bins and m (indistinguishable) balls, how many arrangements are possible such that no bin has greater than r balls?

On Spins and Surprises [10/15/2011]
Nine consecutive spins of a roulette wheel surprise a gambler unfamiliar with how to determine the likelihood of independent events. Doctor Vogler obliges with the requested combinations and probabilities, but only after questioning our arbitrary preference for some events over others that may have the same -- or even longer -- odds.

Onto Functions and Stirling Numbers [09/22/2002]
How would I show for m greater than or equal to 3 that s(m, m-2) = (1/ 24)m(m-1)(m-2)(3m-1), where s(m,n) are Stirling numbers of the first kind?

Optimal Seating Arrangements [07/20/2000]
N people are invited to a party and asked to RSVP with the names of up to k people they would like to sit with. Is there a formula that will yield the "best" arrangement of people?

Ordering Combinations, Then Picking the nth Item [10/22/2010]
How do you pick the nth combination without listing all the ones before it? Doctor Vogler restores some order to an ambiguous objective, then outlines an algorithm for sorting combinatorics.

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