2018 AMC 10B Problem 9

Below is the professionally curated solution for Problem 9 of the 2018 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 10B solutions, or check the answer key.

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Concepts:dice (probability)bijection symmetry

Difficulty rating: 1370

The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6.$ Let $p$ be the probability that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10.$ What other sum occurs with the same probability $p?$

$13$

$26$

$32$

$39$

$42$

Solution:

Replace each die's value $k$ by $7 - k.$ This pairs up outcomes one-to-one and keeps their probabilities, and it sends a total of $s$ to $7 \cdot 7 - s = 49 - s.$ So the sums $s$ and $49 - s$ are equally likely. The partner of $10$ is $49 - 10 = 39.$ Thus, D is the correct answer.

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