Frequency of numbers in Pascal’s triangle


Pascals triangleIn Pascal’s triangle the number 1 appears infnitely many times. The number 2 appears just once. The number 3 appears twice.

Can you prove that all numbers other than 1 will appear a fnite
amount of times?

What is the first number that appears exactly 3 times?

What is the first number that appears exactly 4 times?

Can you find a number that appears exactly 5 times?

Can you find a number that appears exactly 6 times?

Can you find a number that appears more than 6 times?

Reveal Solution
The last row that can contain n is the nth row (counting the initial 1 as the 0th row). n will have appeared a finite amount of times up to this point therefore all numbers other than 1 will appear a finite amount of times.

The first number to appear 3 times is 6, the first number to appear 4 times is 10.

Numbers that appear more than 4 times are much rarer. The smallest is 120 which appears 6 times. The smallest number that appears 8 times is 3003.

No examples have been found of numbers that appear exactly 5 times; however, it is not known whether any exist.

Singmaster’s conjecture states that there is a finite upper bound on the number of times a number can appear. It is thought that this upper bound is 10 or 12, though no such examples have been found therefore it could be as low as 8.

Problem Page Coordinator: Claire Baldwin – Mathematics in Education and Industry
Acknowledgement: The IMA are indebted to MEI for sourcing and supplying Mathematics Today with these well-known puzzles.
First published in Mathematics Today (December 2017)
Published

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