# Tag Archives: combinatorics

## Making Triangles

You have six sticks of lengths , and cm. Find the number of noncongruent triangles that can be formed using three of these sticks as sides. Source: NCTM Mathematics Teacher, August 2006 Solution Three sticks make a triangle. There are … Continue reading

## U.S. Senate Committee

There are members in the U.S. Senate ( from each state). In how many ways can a committee of senators be formed if no state may be represented more than once? Source: NCTM Mathematics Teacher, August 2006 Solution We want … Continue reading

## Five Same Numbers

You roll five fair, six-sided dice, all at the same time. In that one roll, what is the probability of getting five of the same number? Source: NCTM Mathematics Teacher, September 2006 Solution When you roll five dice at the … Continue reading

## Second Number Larger Than The First

Suppose a six-sided die has sides numbered through . If a person throws the die two times, what is the probability that the second number will be larger than the first? Source: NCTM Mathematics Teacher, September 2006 Solution The desirable … Continue reading

## Bowling Pins

After rolling the first ball of a frame in a game of 10-pin bowling, how many different pin configurations can remain (assuming all configurations are physically possible)? Source: NCTM Mathematics Teacher, September 2006 Solution A pin configuration is made up … Continue reading

## Largest of Five Integers

A set contains five integers. When distinct elements of this set are added together, two at a time, the complete list of different possible sums is , and . What is the largest of the five integers in the set? … Continue reading

## Same Difference

If fifteen distinct whole numbers are randomly selected from to inclusive, determine the probability that two pairs of these numbers will have the same difference. Source:NCTM Mathematic Teacher, January 2006 Solution What are the possible difference values? There are possible … Continue reading

## Numbers Greater Than 5,000,000

Using all the digits and , how many distinct integers greater than can be formed? Source: NCTM Mathematics Teacher, January 2006 Solution The first digit must be a or a . Case 1: _ _ _ _ _ _ Since … Continue reading

## Ten Different Digits

What is the probability that a number chosen at random from the range to inclusive will contain ten different digits? Round your answer to the nearest millionth. Source: NCTM Mathematics Teacher, January 2006 Solution We divide the large range into … Continue reading

## Sum of Consecutive Numbers (Part 2)

The sum of consecutive positive integers is . Find all possible values of . Source: NCTM Mathematics Teacher, December 2005 SOLUTION Let be the sum of consecutive positive integers with the first term equal . If we add the first … Continue reading