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The question is "Compute the number of 5-digit integers ABCDE, with all digits distinct, such that the first 3 digits are strictly decreasing, and the last 3 digits are strictly increasing."

This is an old question for an AMC competition since I am practicing for it.

I know that C has to be the smallest number and can only range from 0-5. Can someone solve this for me?

1 Answers1

5

Pick $5$ distinct digits.

The least digit must go in the third position of the number.

Pick two of the remaining $4$ digits to go to the left of the middle digit.

Everything else is forced.

For example if you pick digits 0, 2, 3, 6, 8, then your number must be of the form $**0**$. If you pick $2$ and $6$ to go to the left of the middle digit, your number must be $62038$.

paw88789
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  • So now I just have to find the number of 5 digit numbers with distinct digits and multiply them by 6? –  Jul 09 '20 at 14:23
  • @chem1kal Why don't you try a couple of examples and see if you can answer that question for yourself? It may give you a better feel for the problem. – paw88789 Jul 09 '20 at 14:23
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    Do you know about combinations? That will help you know how many sets of $5$ you can choose from a set of $10$. – paw88789 Jul 09 '20 at 14:25
  • Thank you! I completely forgot about combinations –  Jul 09 '20 at 14:32