Six dice are thrown. The probability that different numbers will turn up is equal to

Options

A. `(5)/(36)`
B. `(5)/(324)`
C. `(3)/(324)`
D. `(1)/(324)`

Answer

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

### Step 1: Calculate the Total Outcomes
When throwing 6 dice, each die can show any number from 1 to 6. Therefore, the total number of outcomes when throwing 6 dice is given by:

\[ \text{Total Outcomes} = 6^6 \]

### Step 2: Calculate the Favorable Outcomes
Now, we need to calculate the number of favorable outcomes where all six dice show different numbers.

1. For the first die, we can have any of the 6 numbers (1 to 6).
2. For the second die, we can have any of the remaining 5 numbers (since it must be different from the first).
3. For the third die, we can have any of the remaining 4 numbers.
4. For the fourth die, we can have any of the remaining 3 numbers.
5. For the fifth die, we can have any of the remaining 2 numbers.
6. For the sixth die, we can only have 1 remaining number.

Thus, the number of favorable outcomes is:

\[ \text{Favorable Outcomes} = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 6! \]

### Step 3: Calculate the Probability
The probability \( P \) that all six dice show different numbers is given by the ratio of favorable outcomes to total outcomes:

\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{6!}{6^6} \]

### Step 4: Simplify the Probability
Now we can calculate \( 6! \) and \( 6^6 \):

- \( 6! = 720 \)
- \( 6^6 = 46656 \)

Thus, the probability becomes:

\[ P = \frac{720}{46656} \]

### Step 5: Simplify the Fraction
To simplify \( \frac{720}{46656} \):

1. Divide both the numerator and the denominator by 720:

\[ P = \frac{1}{64.8} \approx \frac{5}{324} \]

So the final probability that all six dice show different numbers is:

\[ P = \frac{5}{324} \]

### Final Answer
The probability that all six dice show different numbers is \( \frac{5}{324} \).

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