How many natural odd numbers are between 100 and 999 that have all different digits?There are two conditions in this question: (1)the number must be odd and (2)must have all different digits. I thought that condition 1 is more important. So, because tehere is five odd numbers, { 1 , 3 , 5 , 7 , 9 }, the units position has 5 choices. The tens position has 9 choices, since it must be different from the units. The hundreds position has 7 choices, because can't be 0 and must be different from the previous positions.One gets 7 ⋅ 9 ⋅ 5 = 315. But using Excel I found that there are 320 odd numbers with all different digits between 100 and 999. Where I'm wrong? Thanks

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How many natural odd numbers are between 100 and 999 that have all different digits?There are two conditions in this question: (1)the number must be odd and (2)must have all different digits. I thought that condition 1 is more important. So, because tehere is five odd numbers,   {  1  ,  3  ,  5  ,  7  ,  9  }, the units position has 5 choices. The tens position has 9 choices, since it must be different from the units. The hundreds position has 7 choices, because can&#039;t be 0 and must be different from the previous positions.One gets   7  ⋅  9  ⋅  5  =  315. But using Excel I found that there are 320 odd numbers with all different digits between 100 and 999. Where I&#039;m wrong? Thanks

Giovanna Erickson · Accepted Answer

1. 5 ways to choose the last digit2. 8 ways to choose the hundreds digit ( not 0, not last digit chosen )3. 8 ways to choose tens digit ( not either of previous digits )  (  5  )  (  8  )  (  8  )  =  320Always select the restricted digits before the unrestricted ones. We are lucky the restrictions don&#039;t overlap, if they did we would need to break the problem into cases.

How many natural odd numbers are between 100 and 999 that have all different digits? There are

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