A Social Security number has nine digits, using the digits 0 through 9 with repetition of digits allowed. How many Social Security numbers have at least one repeated digit?

Question

anonymous · Answer

We can approach this from the other end. If the SS number were ten digits, then we would know that the numbers 0-9 could be used once
with 10! permutations. We would subtract this from 10^10 possible ten-digit numbers to get those that had one or more repetition. 

If we had only 1-9 to use, we could find the SS repeats similarly from
10^9 - 9!. 

My problem is we have 10 digits for the nine-digit number. Let's use
all but 0, then all but 1, then all but 2, etc. We would have 9! possibilities times each one of these, so 10*9! = 10!. Perhaps.

My best estimate: 10^9 - 10! = #  SS numbers with one or more repetitions.

parthkohli · Answer

Find the number of Social Security numbers that have no repeated digits.

anonymous · Answer

Yes, that's what I attempted. Then the answer to the question would be all possible minus this.

anonymous · Answer

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anonymous · Answer

no digits repeated is (10P9)

so 10^9 - (10P9) = 996371200

anonymous · Answer

@sourwing  Nice to have my guess confirmed. Thanks.

anonymous · Answer

Thank you!