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Math Help - [SOLVED] Permutations and combinations question..

  1. #1
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    [SOLVED] Permutations and combinations question..

    Each of the digits, 1, 1, 2, 3, 3, 4, 6 is written on a separate card. The seven cards are then laid out in a row to form a 7-digit number.

    (a) how many distinct 7-digit numbers are there? [answered - might be related, so posting here]

    7! / 2!x2! = 1260


    (d) How many of these 7-digit numbers start and end with the same digit?

    Do we need to consider distinct objects in this question? I don't get it. I have tried different ways but all failed.
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  2. #2
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    Quote Originally Posted by struck View Post
    Each of the digits, 1, 1, 2, 3, 3, 4, 6 is written on a separate card. The seven cards are then laid out in a row to form a 7-digit number.

    (a) how many distinct 7-digit numbers are there? [answered - might be related, so posting here]

    7! / 2!x2! = 1260


    (d) How many of these 7-digit numbers start and end with the same digit?

    Do we need to consider distinct objects in this question? I don't get it. I have tried different ways but all failed.

    There are two cases : Start with 1 and end with 1 or Start with 3 and end with 3 .

    5!x2
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  3. #3
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    The answer is 120 in the book. Although your answer definitely makes sense.
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  4. #4
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    Hello, struck!

    Each of the digits, 1, 1, 2, 3, 3, 4, 6 is written on a separate card.
    The seven cards are then laid out in a row to form a 7-digit number.

    (a) How many distinct 7-digit numbers are there?
    . . . \frac{7!}{2!\,2!} \:=\:1260 . . . . Right!

    (d) How many of these 7-digit numbers start and end with the same digit?
    You and mathaddict are both correct . . .


    [1] Numbers that begin and end with 1: . 1\:\_\:\_\:\_\:\_\:\_\:1
    . . The other five digits (2,3,3,4,6} can be arranged in: {5\choose2} \,=\,60 ways.

    [2] Numbers that begin and end with 3: . 3\:\_\:\_\:\_\:\_\:\_\:3
    . . The other five digits {1,1,2,4,6} can be arranged in: {5\choose2} \,=\,60 ways.


    Therefore, there are: . 60 + 60 \:=\:{\color{blue}120} numbers that begin and end with the same digit.

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