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Thread: problem related to digits.

  1. #1
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    problem related to digits.

    How many six digit number contains exactly 4 different digit?

    Any help would be greatly appreciated?

    Thanks,
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  2. #2
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    Hello, a69356!

    How many six-digit numbers contains exactly 4 different digits?
    I will assume that the number may begin with zero.


    First, select 4 digits from the available 10 digits.
    . . There are: .$\displaystyle {10\choose4}\:=\:{\color{blue}210}$ choices.


    Then there are two possible distributions of the digits.

    . . . $\displaystyle \begin{array}{cc}(1)& \{A,A,A,B,C,D\} \\ \\[-4mm]
    (2) & \{A,A,B,B,C,D\} \end{array}$



    Case (1): .$\displaystyle A,A,A,B,C,D$
    . . We have a Triple and three Singletons.

    There are 4 choices for the Triple

    Then the letters can be arranged in: $\displaystyle {6\choose3,1,1,1} = {\color{blue}120} $ ways.

    There are: .$\displaystyle 210\cdot4\cdot120 \:=\:110,\!800\text{ Case-one numbers.}$



    Case (2): .$\displaystyle A,A,B,B,C,D$
    . . We have two Pairs and two Singletons.

    There are: $\displaystyle {4\choose2} \:=\:{\color{blue}6}$ choices for the two Pairs.

    Then the letters can be arranged in: $\displaystyle {6\choose2,2,1,1} \:=\:{\color{blue}180}$ ways.

    There are: .$\displaystyle 210\cdot6\cdot180 \:=\:226,800 \text{ Case-two numbers.}$



    Therefore, there are: .$\displaystyle 100,\!800 + 226,\!800 \;=\;\boxed{327,\!600}$ six-digit numbers
    . . that contain four different digits.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    If leading zeros are not allowed, consider this fact:
    . . One-tenth of the above numbers begin with zero.

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  3. #3
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    Thanks a lot Soroban it is really helpful.

    again thanks
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