Probability & Number Theory

Number Theory Level 3

If we randomly construct a nine-digit number using each of the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once, what is the probability that it is divisible by 6?


The answer is 0.44444444.

H K by H K , 18, France

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1 solution

H K
Mar 6, 2017

A number that is divisible by 6 has to be even and divisible by 3. Since all numbers that can be constructed have digit sum 45, they are all multiples of 3. Hence, it is divisible by 6 if and only if the last digit is even (such that the number is a multiple of 2). This occurs with probability 4/9 .

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How would the answer change if we allowed for the digit 0, and wanted a 10-digit number?

Calvin Lin Staff - 4 years, 3 months ago

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We will restrict the 0 from entering the 1st place and then do.. isnt it Calvin sir?

Md Zuhair - 4 years, 3 months ago

Also this is discrete mathematics problem . No way it is number theory

Md Zuhair - 4 years, 3 months ago

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Since the crucial part is "How do we count the number of positive cases", as opposed to "Oh, we identified that probability = positive cases / total cases", I have placed this in Number Theory.

It's like saying "Hey, he used 1 + 2 + ... + 9 = 45 , so that's basic math" and hence arguing that this is a problem in basic math. Yes, that concept is needed, but it's not truly the crux of the problem.

Calvin Lin Staff - 4 years, 3 months ago

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Sir can you reveiw my Wiki page here https://brilliant.org/wiki/rank-of-a-word-in-dictionary/

Md Zuhair - 4 years, 3 months ago

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