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If there is a ten digit number (like a US 10 digit telephone number) such as 7177425231, how many different ways can the digits be rearranged? I understand how you would go about finding this if you could only use the digits 0-9 once in the ten digit number, but since this example has some digits out of 0-9 which are missing, and some digits which are repeated (like 7) I'm not sure how you would go about doing this?

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In addition to the answer you received, you might want to review this entire paper and do all the problems, especially problem 1.18. newagepublishers.com/samplechapter/001488.pdf –  Amzoti Dec 8 '12 at 21:57

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The locations for the $7$'s can be chosen in $\dbinom{10}{3}$ ways. For each such choice, the locations of the $2$'s can be chosen in $\dbinom{7}{2}$ ways. Once this is done, the locations of the $1$'s can be chosen in $\dbinom{5}{2}$ ways. The singletons can then be arranged in $3!$ ways. Multiply.

There is nothing special about the order in which this was done. We could say that the location of the $3$ can be any of $10$ places, and then the location of the $4$ can be any of $9$, and then the location of the $5$ any of $8$. Then the locations of the $1$'s can be chosen in $\dbinom{7}{2}$ ways, and the locations of the $2$'s in $\dbinom{5}{2}$ ways. The $7$'s now must go into the empty places.

Another way: Paint the repeated numbers so as to make them distinguishable. There are $10!$ ways to arrange the new version of the digits. Now unpaint the $7$'s. Always $3!$ of the old painted arrangements collapse into one. So we need to divide by $3!$. Unpaint the $1$'s, and the $2$'s. That divides the number by another $2!2!$.

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Ah, this makes much more sense now! And your alternate method was especially helpful for a visual learner :) –  verovdp Dec 8 '12 at 22:05

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