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

• A. \( \frac{1}{324} \)
• B. \( \frac{5}{324} \)
• C. \( \frac{7}{324} \)
• D. \( \frac{11}{324} \)

Hint:

When calculating probabilities involving permutations, always account for the order of outcomes when the arrangement matters.

Answer 1

The Correct Option is B

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: \[ \boxed{\frac{5}{324}} \]