Difference between revisions of "2021 April MIMC 10 Problems/Problem 15"
Cellsecret (talk | contribs) (Created page with "Paul wrote all positive integers that's less than <math>2021</math> and wrote their base <math>4</math> representation. He randomly choose a number out the list. Paul insist t...") |
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<math>\textbf{(A)} ~30 \qquad\textbf{(B)} ~62 \qquad\textbf{(C)} ~64 \qquad\textbf{(D)} ~84 \qquad\textbf{(E)} ~126</math> | <math>\textbf{(A)} ~30 \qquad\textbf{(B)} ~62 \qquad\textbf{(C)} ~64 \qquad\textbf{(D)} ~84 \qquad\textbf{(E)} ~126</math> | ||
==Solution== | ==Solution== | ||
| − | + | First, we can convert <math>2021</math> to base <math>4</math>. <math>2021_10=133211_4</math>. Therefore, the total ways to obtain only <math>2</math> and <math>3</math> as its digits that are less than <math>2^5+2^4+2^3+2^2+2^1+2^0=2^6-2=\fbox{\textbf{(B)} 62}</math>. | |
Latest revision as of 13:44, 26 April 2021
Paul wrote all positive integers that's less than 
and wrote their base 
representation. He randomly choose a number out the list. Paul insist that he want to choose a number that had only 
and 
as its digits, otherwise he will be depressed and relinquishes to do homework. How many numbers can he choose so that he can finish his homework?
Solution
First, we can convert to base . 
. Therefore, the total ways to obtain only and as its digits that are less than 
.
