2021 JMC 10 Problems/Problem 4

Problem

A day in the format $mm/dd$ is called binary if all of the digits are either $0$ s or $1$ s with leading zeros allowed. How many days in a year are binary?

$\textbf{(A) } 5 \qquad\textbf{(B) } 9 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 15 \qquad\textbf{(E) } 16$

Solution

The month $mm$ and date $dd$ cannot be of the form $00$ . Each digit $m$ and $d$ can be either $0$ or $1,$ but we must subtract the undesired case of $00.$ So, $mm$ only has $2^2 - 1 = 3$ possibilities, and $dd$ also has $2^2 - 1 = 3$ possibilities. In total, there are $3^2 = 9$ binary days in a year.

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2021 JMC 10 Problems/Problem 4

Problem

Solution