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

Options

A. `(1)/(6^(6))`
B. `(1)/(6^(5))`
C. `(1)/(6)`
D. none of these

Explanation

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

### Step 1: Determine the total number of outcomes
When throwing a single die, there are 6 possible outcomes (1 through 6). Since we are throwing 6 dice simultaneously, the total number of outcomes is given by:
\[ \text{Total Outcomes} = 6^6 \]

### Step 2: Determine the number of favorable outcomes
For all six dice to show the same face, they can show any one of the 6 faces (1, 2, 3, 4, 5, or 6). Therefore, the number of favorable outcomes (where all dice show the same face) is:
\[ \text{Favorable Outcomes} = 6 \]

### Step 3: Calculate the probability
The probability \( P \) that all six dice show the same face is calculated using the formula:
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{6}{6^6} \]

### Step 4: Simplify the probability
We can simplify the probability:
\[ P = \frac{6}{6^6} = \frac{1}{6^5} \]

### Final Answer
Thus, the probability that all six dice show the same face is:
\[ P = \frac{1}{7776} \]