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Browse High School Permutations/Combinations

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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 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...

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.

Transporting Twenty in Two Ways, and the Twins Together [04/16/2014]
An ambiguous combinatorics question confounds a student and his classmates — along with the teacher who made it up. Doctor Peterson troubleshoots the problem's possible cases, and further sets up and solves a simpler problem, emphasizing throughout the importance of precise wording.

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...

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