2021 AMC 10 A Fall

Complete problem set with solutions and individual problem pages

Problem 13 Medium

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?(2021 AMC Fall 10A, Question #13)

  • A.

    \frac{1}{64}

  • B.

    \frac{1}{6}

  • C.

    \frac{1}{4}

  • D.

    \frac{5}{16}

  • E.

    \frac{1}{2}

Answer:D

Solution 1:

Note that for this restriction to be true, there must be 3 balls of each color. There are a total of 2^{6}=64 ways to color the balls, and there are \left(\begin{array}{l}6 \\ 3\end{array}\right)=20 ways for three balls chosen to be painted white. Thus, the answer is \frac{20}{64}=(\text{D}) \frac{5}{16}

Solution 2:

For this restriction to be upheld, there must be three black and three white balls. One such way for this to occur is the arrangement B B B W W W W, which has a \frac{1}{2^{6}} probability of occuring. However, there are \frac{6 !}{3 ! \cdot 3 !} ways to arrange the three black and three white balls, meaning that the answer is, \frac{1}{64} \cdot \frac{6 !}{3 ! \cdot 3 !}= (D) \frac{5}{16}

Related Problems
Problem 11 2021 AMC 10 A Fall
Problem 12 2021 AMC 10 A Fall
Problem 14 2021 AMC 10 A Fall
Problem 15 2021 AMC 10 A Fall
Think Academy Powered by ChatGPT