Using the numbers 0,1,2,...,9 as digits, how many 4-digit numbers exist in which all digit are different with two digits as even numbers and two as odd numbers?

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Daniel Valdez · Accepted Answer

First pick 2 odd numbers and 2 even numbers. There are                           (                    5        2                          )              2   ways to do this. Now order them. There are 4! ways. So there are   4  !                          (                    5        2                          )              2   total numbers, but we have included 4-digit numbers that start with 0. Now let&#039;s count the number that start with 0. There are 4 choices for the other even digit,                 (              5      2                  )       choices for the odd digits, and 3! orders. So for our final answer we have:  4  !                          (                    5        2                          )              2    −  3  !  (  4  )                (              5      2                  )

Arraryeldergox2 · Accepted Answer

Another possibility would be this : consider the 4-digit numbers without 0 in them. There is 4 possible even numbers and 5 possible odd numbers, so                 (              5      2                  )                      (              4      2                  )        =  60 possibilities. There are   4  !  =  24 ways to place them, hence you have   60  ×  24  =  1440 such numbers.Now consider those with 0 in them. The zero must be one of the 3 first digits. Now for the number possibilities, there are 4 choices for the even number and                 (              5      2                  )        =  10  choices for the odd numbers, hence 40 choices for the numbers. Afterwards, there are 3 possibilities for the position of 0 and   3  !  =  6 choices for the position of the three other digits, hence 18 possibilities in total. Therefore there is   40  ×  18  =  720 such digits.Summing up gives   1440  +  720  =  2160  =  4  !                          (                    5        2                          )              2    −  3  !  (  4  )                (              5      2                  )       possibilities.

Using the numbers 0,1,2,...,9 as digits, how many 4-digit numbers

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