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Difference between revisions of "2021 JMC 10 Problems/Problem 4"

(Created page with "==Problem== A day in the format <math>mm/dd</math> is called ''binary'' if all of the digits are either <math>0</math>s or <math>1</math>s with leading zeros allowed. How many...")
 
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Latest revision as of 16:07, 1 April 2021

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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