# Frequency of numbers in Pascal’s triangle

In 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.

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