A good password consists of 10 characters containing lowercase letters,uppercase letters, and numbers. Find the total number of good passwords. Note that an uppercase letter is different from a lowercase letter.
2017 MTG Sharp Math Trail Problem
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$\begingroup$
combinatorics
permutations
inclusion-exclusion
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0You surely mean ***containing at least one of each***, but at least to me the statement is not crystal clear about it. – 2017-02-04
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1Please do not *command* things. What have you tried? – 2017-02-04
1 Answers
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Use inclusion-exclusion.
First write the number of passwords made of (I use shorthands) UL, LL and NU. This is $$(26 + 26 + 10)^{10}.$$
Then subtract the number of passwords made only of UL and LL, the number of passwords made only of UL and NU, the number of passwords made only of LL and NU.
But by doing this, you have subtracted twice the number of passwords made only of LL, the number of passwords made only of UL, the number of passwords made only of NU. So add them back.