Six dice are thrown. The probability that different number will turn up is

Options

A. `129/1296`
B. `1/54`
C. `5/324`
D. `5/54`

Answer

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

### Step 1: Determine the Total Outcomes
When throwing a single die, there are 6 possible outcomes (1 through 6). When throwing 6 dice, the total number of outcomes is calculated as:
\[ \text{Total Outcomes} = 6^6 \]
This is because each die is independent, and each can land on any of the 6 faces.

### Step 2: Determine the Favorable Outcomes
To find the favorable outcomes where all six dice show different numbers, we need to arrange the numbers 1 through 6. The number of ways to arrange 6 distinct numbers is given by the factorial of 6:
\[ \text{Favorable Outcomes} = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

### Step 3: Calculate the Probability
The probability is calculated using the formula:
\[ \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{6!}{6^6} \]

### Step 4: Simplify the Probability
Now we will simplify the expression:
\[ \text{Probability} = \frac{720}{6^6} \]
Calculating \(6^6\):
\[ 6^6 = 46656 \]
Thus, the probability becomes:
\[ \text{Probability} = \frac{720}{46656} \]

### Step 5: Reduce the Fraction
To simplify \(\frac{720}{46656}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD). After performing the division, we find:
\[ \frac{720 \div 144}{46656 \div 144} = \frac{5}{324} \]

### Final Answer
Thus, the probability that all six dice show different numbers is:
\[ \frac{5}{324} \]