Three standard dice (each marked with faces numbered 1 through 6) are rolled. How many outcome combinations exist where at least one die displays the digit 2?
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Answer: C. 91
Explanation
Rolling three standard dice generates a total sample space of two hundred sixteen possible outcomes. Calculating outcomes containing no digit two involves cubing five non matching faces to get one twenty five. Subtracting the outcomes with no twos from the total sample space leaves ninety one combinations. Probability and combinatorics problems can often be solved by counting the complementary undesirable outcomes. Each standard die has six distinct faces numbered one through six in traditional games. The complement rule states that the probability of an event equals one minus the probability of its opposite.
Combinatorics principles are essential for calculating probabilities in tabletop board games and gambling. Statistical analysis utilizes counting techniques to determine likelihoods in complex sample spaces. Computer algorithms generate random test cases by evaluating combinations of input parameters. Option A suggests thirty six which drastically underestimates the number of combinations. Option B proposes eighty one which results from an incorrect calculation of the complementary outcomes. Option D offers one hundred sixteen which overestimates the number of valid target outcomes.
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