Six dice are thrown simultaneously. The probability that all of them show the different faces, is

Options

A. `(1)/(6^(5))`
B. `(6!)/(6^(6))`
C. `(1)/(6!)`
D. `(5!)/(6!)`

Explanation

To find the probability that all six dice show different faces when thrown simultaneously, we can follow these steps:

### Step-by-Step Solution:

1. **Total Possible Outcomes**:
When throwing 6 dice, each die has 6 faces. Therefore, the total number of possible outcomes when throwing 6 dice is:
\[ 6^6 \]
This is because each die can land on any of the 6 faces independently.

2. **Favorable Outcomes (Different Faces)**:
To find the number of favorable outcomes where all dice show different faces, we can think of it as arranging 6 different faces on the 6 dice.
- For the first die, we can have any of the 6 faces.
- For the second die, we can have any of the remaining 5 faces.
- For the third die, we can have any of the remaining 4 faces.
- For the fourth die, we can have any of the remaining 3 faces.
- For the fifth die, we can have any of the remaining 2 faces.
- For the sixth die, we can have the last remaining face.

Thus, the number of ways to have all different faces is:
\[ 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 6! \]

3. **Calculating the Probability**:
The probability \( P \) that all six dice show different faces is given by the ratio of the number of favorable outcomes to the total possible outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{6!}{6^6} \]

4. **Final Calculation**:
We can calculate \( 6! \) and \( 6^6 \):
- \( 6! = 720 \)
- \( 6^6 = 46656 \)

Therefore, the probability is:
\[ P = \frac{720}{46656} \]

5. **Simplifying the Probability**:
We can simplify this fraction:
\[ P = \frac{720 \div 720}{46656 \div 720} = \frac{1}{64.8} \approx 0.0154321 \]

### Final Answer:
Thus, the probability that all six dice show different faces is:
\[ P = \frac{720}{46656} \approx 0.0154321 \]