Question:
When six unbiased dice are rolled simultaneously, the probability of getting all distinct numbers (i.e., \( 1, 2, 3, 4, 5, \) and \( 6 \)) is
When six unbiased dice are rolled simultaneously, the probability of getting all distinct numbers (i.e., \( 1, 2, 3, 4, 5, \) and \( 6 \)) is
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When calculating probabilities involving permutations, always account for the order of outcomes when the arrangement matters.
Updated On: Jan 23, 2025
- \( \frac{1}{324} \)
- \( \frac{5}{324} \)
- \( \frac{7}{324} \)
- \( \frac{11}{324} \)
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The Correct Option is B
Solution and Explanation
To find the probability of rolling six distinct numbers, calculate the total number of favorable outcomes and divide by the total number of possible outcomes.
Step 1: Total possible outcomes.
\[
6^6 = 46656
\]
Step 2: Favorable outcomes.
The first die can take any of the 6 numbers, the second can take any of the remaining 5, and so on:
\[
6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720
\]
Step 3: Calculate the probability.
\[
P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{720}{46656} = \frac{5}{324}
\]
Final Answer:
\[
\boxed{\frac{5}{324}}
\]
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