Five dice are tossed. What is the probability that the five numbers shown will be different?

Options

A. `(5)/(54)`
B. `(5)/(18)`
C. `(5)/(27)`
D. `(5)/(81)`

Answer

To find the probability that all five numbers shown on the tossed dice are different, we can follow these steps:

### Step 1: Calculate the Total Outcomes
When tossing one die, there are 6 possible outcomes (1 through 6). For five dice, the total number of outcomes is calculated as:
\[ \text{Total Outcomes} = 6^5 \]
This is because each die operates independently, and each has 6 outcomes.

### Step 2: Calculate the Favorable Outcomes
To find the number of favorable outcomes where all five dice show different numbers, we can think through the process of rolling the dice:

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

Thus, the number of favorable outcomes can be calculated as:
\[ \text{Favorable Outcomes} = 6 \times 5 \times 4 \times 3 \times 2 = 6! \]

### Step 3: Calculate the Probability
The probability \( P \) that all five numbers shown will be different is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{6!}{6^5} \]

### Step 4: Simplify the Probability
Calculating \( 6! \) gives us:
\[ 6! = 720 \]
And \( 6^5 \) gives us:
\[ 6^5 = 7776 \]
Thus, the probability can be simplified as:
\[ P = \frac{720}{7776} \]
Now, we can simplify this fraction. Dividing both the numerator and the denominator by 720 gives us:
\[ P = \frac{1}{10.8} = \frac{5}{54} \]

### Final Answer
The probability that all five numbers shown will be different is:
\[ \frac{5}{54} \]

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