Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I want to check how many integral numbers in $\big[1,10^6\big]$ include the numbers $1,2,3,4,5$ and how many only them.
how should I check it? this is a problem of inclusion-exclusion?
I would like to get some advice!

share|cite|improve this question
I'm guessing you want integers, but as it is written, there are uncountably many since you can count decimal points. – Clayton Aug 24 '13 at 12:23
I need Integers.. – Ofir Attia Aug 24 '13 at 12:39
I understand this problem as this: How many integers exist in the inclusive interval $[1..10^6]$ that a) have at least one occurrence of one of the numbers [1,2,3,4,5] b) are entirely composed of the numbers [1,2,3,4,5]. Is that correct? – Murch Aug 24 '13 at 12:44
Yes, I think about problem that I have 7 places to set the numbers [1,2,3,4,5] on them, but I dont know if its right – Ofir Attia Aug 24 '13 at 12:45

Exclude $10^6$, and consider $[0,999999]$.

For your first question:

There are $10^6$ numbers in the interval $[0,999999]$. Each number in this interval can be thought of as having $6$ digits (So, $27$ would be $000027$). The numbers that do not satisfy the given property are entirely composed of $0,6,7,8,9$. So the total number of numbers to be excluded are $5^6$.

For your second question:

There are exactly $5$ one-digit numbers that satisfy this property. Similarly, there are $5^2$ two-digit numbers with this property. Continue and then sum all the numbers.

Make sure to consider $10^6$ in the end.

share|cite|improve this answer


  • You do not need use inclusion-exclusion, though you can if you want to. Another approach to the first question could be to look at those which do not include any of $1,2,3,4,5$

  • You might find it easier to treat $1000000$ and $0$ as special cases and instead look at the first question as if it was asking you to look at $[0,999999]$ with an adjustment.

  • For the second question you might split the range by numbers of digits, i.e. into $[1,9]$, $[10,99]$, etc.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.