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How many numbers are greater than 250, use only digits 1,2,3,4,5 and all their digits are different?

When I approached this question, I tried to do the relevant number of options for 3 digits numbers + 4 digit numbers +..., and this actually going to an infinite number. Is this the right way to solve it?

YuiTo Cheng
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M.Mitelman
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    Really? Can you tell me a six digit number using only digits $1,2,3,4,5$ with all digits different? Any one will do. Your approach is OK in terms of break up of digits, but do go into more detail regarding each "number of digits is $k$" case more carefully. – Sarvesh Ravichandran Iyer Mar 28 '19 at 08:16
  • Oh , I didn't thought about it, thank you! – M.Mitelman Mar 28 '19 at 08:23
  • No issues, just to confirm finiteness. Now do as the answer below directs. Essentially, employ free choice for four and five digit numbers, and for three digit number you will have to be careful about the first digit and carefree about the rest! – Sarvesh Ravichandran Iyer Mar 28 '19 at 08:30
  • @астон вілла олоф мэллбэрг almost carefree about the rest. If the first digit is $2$, then you have to be careful about what your second digit is. – 5xum Mar 28 '19 at 08:34
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    @5xum True, I was carefree(or careless, more like) with the word carefree there! – Sarvesh Ravichandran Iyer Mar 28 '19 at 08:35
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    @астонвіллаолофмэллбэрг I solved the question, and the result I got is 279 optional numbers - 39 of 3 digit numbers , 120 of 4 digit numbers , and 5! of 5 digit numbers. – M.Mitelman Mar 28 '19 at 08:46
  • The four and five digit calculations are correct. For the three digit calculation, I think you are correct as well : starting with each of $3,4,5$ there are $4 \times 3 = 12$ numbers, and then you add to this $251,253,254$ which gives $39$. Yes, I think you are correct. – Sarvesh Ravichandran Iyer Mar 28 '19 at 08:54

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The number is not going to be infinite. For example, there is no $6$ digit number that uses only $1,2,3,4,5$ and has all digits different. Therefore, you only need to check the number of $3$ digit, $4$ digit and $5$ digit numbers that fit your condition.

It should be easy to see that for $4$ and $5$ digit numbers, all of them are larger than $250$, so the task should be quite easy there. For $3$ digit numbers, you might want to split the cases when the number starts with $2$ or if it starts with $3,4$ or $5$ (since it cannot start with $1$ and be above $250$).

5xum
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