Difference between revisions of "2021 Fall AMC 10A Problems/Problem 13"
Aidensharp (talk | contribs) (Created page with "==Problem== Each of <math>6</math> balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is differe...") |
Aidensharp (talk | contribs) (→Solution) |
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==Solution== | ==Solution== | ||
Note that for this restriction to be true, there must be <math>3</math> balls of each color. There are a total of <math>2^6 = 64</math> ways to color the balls, and there are <math>{6 \choose 3} = 20</math> ways for three balls chosen to be painted white. Thus, the answer is <math>\frac{20}{64} = \boxed{\textbf{(D) } \frac{5}{16}}</math>. | Note that for this restriction to be true, there must be <math>3</math> balls of each color. There are a total of <math>2^6 = 64</math> ways to color the balls, and there are <math>{6 \choose 3} = 20</math> ways for three balls chosen to be painted white. Thus, the answer is <math>\frac{20}{64} = \boxed{\textbf{(D) } \frac{5}{16}}</math>. | ||
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| + | -Aidensharp | ||
Revision as of 20:00, 22 November 2021
Problem
Each of 
balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other 
balls?
Solution
Note that for this restriction to be true, there must be 
balls of each color. There are a total of 
ways to color the balls, and there are 
ways for three balls chosen to be painted white. Thus, the answer is 
.
-Aidensharp
