Four dice are thrown simultaneously . Find the probability that :
All of them show the different face .

Options

A. `(1)/( 216)`
B. `(5)/( 18)`
C. `(15)/(36)`
D. `(11)/(36 )`

Answer

To solve the problem of finding the probability that all four dice show different faces when thrown simultaneously, we can follow these steps:

### Step 1: Determine the Total Outcomes
When four dice are thrown, each die has 6 faces. Therefore, the total number of outcomes in the sample space can be calculated as:
\[ \text{Total Outcomes} = 6^4 \]
Calculating this gives:
\[ 6^4 = 1296 \]

### Step 2: Determine the Favorable Outcomes
Next, we need to find the number of favorable outcomes where all four dice show different faces.

1. For the first die, we can choose any of the 6 faces.
2. For the second die, we can choose from the remaining 5 faces (since it must be different from the first).
3. For the third die, we can choose from the remaining 4 faces.
4. For the fourth die, we can choose from the remaining 3 faces.

Thus, the number of favorable outcomes is calculated as:
\[ \text{Favorable Outcomes} = 6 \times 5 \times 4 \times 3 \]
Calculating this gives:
\[ 6 \times 5 = 30 \\ 30 \times 4 = 120 \\ 120 \times 3 = 360 \]

### Step 3: Calculate the Probability
The probability \( P \) that all four dice show different faces is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{360}{1296} \]

### Step 4: Simplify the Probability
To simplify \( \frac{360}{1296} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 360 and 1296 is 72. Therefore:
\[ P = \frac{360 \div 72}{1296 \div 72} = \frac{5}{18} \]

### Final Answer
Thus, the probability that all four dice show different faces is:
\[ \boxed{\frac{5}{18}} \]

---