2021 AMC 10A Fall Problem 13

Below is the professionally curated solution for Problem 13 of the 2021 AMC 10A Fall, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 10A Fall solutions, or check the answer key.

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Concepts:basic probability combinations

Difficulty rating: 900

13.

Each of $6$ 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 $5$ balls?

$\dfrac{1}{64}$

$\dfrac{1}{6}$

$\dfrac{1}{4}$

$\dfrac{5}{16}$

$\dfrac{1}{2}$

Solution:

Note that for this restriction to hold, there must be $3$ balls of each color.

There are $2^6 = 64$ ways to color the balls and $\binom{6}{3} = 20$ to choose which balls are white.

The desired probability is therefore $\dfrac{20}{64} = \dfrac{5}{16}.$

Thus, D is the correct answer.

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