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Math Help - Counting-Permutations

  1. #1
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    Counting-Permutations

    I have a question asking me to find:
    The sum of all 4 digit numbers containing the digits 2,4,6,8 without repetitions

    I realise that the digits can be chosen in (4)_4=4!=24 ways since repetition is not permitted and order does matter

    The final answer is 133320

    Any help would be great
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  2. #2
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    Let the 24 numbers be a_1,\dots,a_{24}. Further, let the ith number a_i be 1000b_{i1}+100b_{i2}+10b_{i3}+b_{i4} where all b_{ij} are in {2,4,6,8}. Let's find S_1=\sum_{i=1}^{24}b_{i1}. Each of the four numbers is encountered 6 times, so the S_1 = 6(2 + 4 + 6 + 8) = 120. Therefore, the first digits of all numbers contribute 120 * 1000 to the whole sum. Similarly, the second digits contribute 120 * 100 and so on.
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  3. #3
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    Quote Originally Posted by qwerty10 View Post
    I have a question asking me to find:
    The sum of all 4 digit numbers containing the digits 2,4,6,8 without repetitions
    The final answer is 133320
    Think about it, each of those four numbers will be in each decimal place six times. The ones column adds up to 6(2+4+6+8)=120.
    Check out \displaystyle120\left( {\sum\limits_{k = 0}^3 {10^k } } \right) = ~?.
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  4. #4
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    Hello, qwerty10!

    Find the sum of all 4 digit numbers containing the digits 2,4,6,8 without repetitions.

    The answer is: 133,320.

    List the 4! = 24 permutations of the four digits
    ,. . and consider their sum.


    . . \begin{array}{cc}<br />
&2468 \\ &2486 \\ &2648 \\ &2684 \\ &2846 \\ &2864 \\ &\vdots \\<br />
&8246 \\ &8264 \\ &8426 \\ &8462 \\ &8624 \\ + & 8642 \\ \hline <br />
\end{array}



    We find that each column has: six 2's, six 4's, six 6's, six 8's.

    The total of each column is: . 6\!\cdot\!2 + 6\!\cdot\!4 + 6\!\cdot\!6 + 6\!\cdot\!8 \:=\:120


    Hence, the addition has the form:

    . . \begin{array}{cccccc}<br />
&&& 1 & 2 & 0 \\ && 1 & 2 & 0 \\ & 1 & 2 & 0 \\ 1 & 2 & 0 \\ \hline 1 & 3 & 3 & 3 & 2 & 0 \end{array}


    Therefore, the sum is: . 133,\!320



    Edit: Plato beat me to it . . . *sigh*
    .
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