Tag Archives: combinatorics

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

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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

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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

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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

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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

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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

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Different Paths

We must travel from to along the lines shown. If we only allow movement upward or to the right, how many different paths are there from to ? Source: NCTM Mathematics Teacher, December 2005 SOLUTION If we fill in the … Continue reading

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Planting Flowers

Four triangular gardening plots form a square as shown. Each plot will contain one kind of flower, and flowers in plots that share an edge will be different. How many different ways can the garden be planted if the flowers … Continue reading

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Painting Train Cars

Each car of a five-car train must be painted a solid color. The only color choices are red, blue, and yellow. If each of these colors must be used for at least one car, in how many ways can this … Continue reading

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Divisible by 25

How many 5-digit numbers with all digits nonzero and no digit repeated are divisible by ? Source: NCTM Mathematics Teacher, November 2006 SOLUTION A number is divisible by if it ends in , or . The nonzero digit condition restricts … Continue reading

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