2018 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 3 Easy

A unit of blood expires after 10!=10\cdot 9\cdot 8\cdots 1 seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire? (2018 AMC 10A Problem, Question#3)

  • A.

    January 2

  • B.

    January 12

  • C.

    January 22

  • D.

    February 11

  • E.

    February 12

Answer:E

There are 10! seconds that the blood has before expiring and there are 60\cdot 60\cdot 24 seconds in a day. Dividing \frac{10!}{60\cdot 60\cdot 24} gives 42 days. 42 days after January 1 is \boxed{(\rm E)~February~12}.

The problem says there are 10!=10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1 seconds. Convert 10! seconds to minutes by dividing by 60: 9\cdot 8\cdot 7\cdot 5\cdot 4\cdot 3\cdot 2 minutes. Convert minutes to hours by again, dividing by 60: 9\cdot 8\cdot 7\cdot 2 hours. Convert hours to days by dividing by 24: 3\cdot 7\cdot 2=42 days.

Now we need to count 42 days after January 1. Since we start on Jan. 1, then we can't count that as a day itself. When we reach Jan. 31(The end of the month), we only have counted 30 days. 42-30=12 . Count 12 more days, resulting \boxed{(\rm E)~February~12}.

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