AMC 8 Daily Practice Round 11
Complete problem set with solutions and individual problem pages
Vivi forgot the password to her safe, but she remembers that it consists of four nonzero digits, and the sum of the digits is . What is the minimum number of attempts needed to guarantee opening the safe?
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To find all possible passwords, we need to determine combinations of four nonzero digits that sum to . The valid combinations are ; ; ; ; ; and , resulting in six groups. For the combination , the digit can occupy any of the 4 positions, giving possibilities. For , can occupy 4 positions, can occupy 3 remaining positions, and fills the rest, yielding possibilities. Similarly, each of , , and also has possibilities. For , can occupy any of the 4 positions, giving possibilities. By the addition principle, the total number of passwords is:
Thus, Vivi needs to try at least attempts to guarantee opening the safe.
