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Having only digits $1,2,3$. How many $10$-digit numbers can you make with these digits such that you do not use 1 at all? ($2^{10}$?)

How many $10$-digit numbers can you make with these digits provided you use $1$ twice? ($2^8$?)

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    Why do you think it is $2^{10}? What led you to guess that (what if all three were allowed)? In the second problem, is it exactly two 1's or at least two 1's (your answer will obviously be different depending)?2011-04-10

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