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In a city with telephone numbers with $6$ digits,how many telephone numbers exist without repeated digits?What does this mean? That my telephone number shouldn't have twice the same number or that my telephone number and yours should not have the same digits?

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    $\begingroup$ Pay attention to the difference between a digit, and a number. A digit is a single character, but a number can consist of one or several digits. $\endgroup$
    – Alec
    Commented Apr 25, 2014 at 12:29
  • $\begingroup$ I understand. Thanks! $\endgroup$
    – Mary Star
    Commented Apr 25, 2014 at 12:38

2 Answers 2

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It means that 122345 is not allowed, nor is 123452, because each of them includes the digit 2 twice. Only numbers like 125783 are allowed.

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  • $\begingroup$ So is the number of the different telephone numbers $$9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4=\frac{9!}{3!}$$ ? $\endgroup$
    – Mary Star
    Commented Apr 25, 2014 at 12:28
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    $\begingroup$ No. You're not considering 0 to be a digit. $\endgroup$
    – John Bupit
    Commented Apr 25, 2014 at 12:31
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It means that each telephone number consists of 6 distinct digits. You need to find all such telephone numbers.

For example, $125432$ isn't a valid telephone number because $2$ appears twice, however, $125430$ is valid.

Assuming that $0$ cannot be the first digit, the number of distinct telephone numbers would be:

$$ 9.9.8.7.6.5 = \frac{9.9!}{4!} $$

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  • $\begingroup$ Ok,I understand..Thank you for your answer!!! $\endgroup$
    – Mary Star
    Commented Apr 25, 2014 at 12:37

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