In how many ways can we list out a string of seven numbers (using only the numbers 1, 2, and 3) such that they sum to 10?

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Josie123 · Accepted Answer

Step 1Let&#039;s first look at the number of ways we can have 7 digits with the number 1, 2, and 3 add up to 10:      3    +    2    +    1    +    1    +    1    +    1    +    1    =    10        2    +    2    +    2    +    1    +    1    +    1    +    1    =    10  In fact, there are no other combinations of numbers that will get us to 10.Ok, so how many permutations can we make with these combinations of numbers?Let&#039;s do       3    +    2    +    1    +    1    +    1    +    1    +    1    =    10   first:There are 7 places the number 3 can go - so that is 7.Once the 3 is placed, there are 6 places the 2 can go - so that is 6.The rest of the numbers are 1&#039;s and so there is only 1 way to do the &quot;filler&quot; with the 1s.So there are       7    ×    6    =    42   different numbers we can make with this combination.Step 2Now to       2    +    2    +    2    +    1    +    1    +    1    +    1    =    10  We can place the three 2s across the seven places in many different ways. This is a combinations problem (we don&#039;t care which 2 ends up where), and so there are:            C              7        ,        3              =                  7        !                              (          3          !          )                          (          7          -          3          )                !              =                  7        !                              (          3          !          )                          (          4          !          )                     ways, which equals:                    7        ×                  6                ×        5        ×                              4            !                                                            3            ×            2                          ×                              4            !                                =    35  And so there are       42    +    35    =    77   different numbers that can be made.

In how many ways can we list out a string

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