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. `(8)/(81)`

Answer

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

### Step 1: Determine the Total Sample Space
When we toss 5 dice, each die has 6 faces. Therefore, the total number of outcomes when tossing 5 dice is given by:
\[ \text{Total Sample Space} = 6^5 \]

### Step 2: Calculate the Total Sample Space
Calculating \(6^5\):
\[ 6^5 = 7776 \]

### Step 3: Determine the Favorable Outcomes
Next, we need to find the number of ways to get 5 different numbers when tossing 5 dice.

1. For the first die, we can choose any of the 6 numbers.
2. For the second die, we can choose from the remaining 5 numbers.
3. For the third die, we can choose from the remaining 4 numbers.
4. For the fourth die, we can choose from the remaining 3 numbers.
5. For the fifth die, we can choose from the remaining 2 numbers.

Thus, the number of favorable outcomes is:
\[ \text{Favorable Outcomes} = 6 \times 5 \times 4 \times 3 \times 2 \]

### Step 4: Calculate the Favorable Outcomes
Calculating \(6 \times 5 \times 4 \times 3 \times 2\):
\[ 6 \times 5 = 30 \\ 30 \times 4 = 120 \\ 120 \times 3 = 360 \\ 360 \times 2 = 720 \]
So, the number of favorable outcomes is 720.

### Step 5: Calculate the Probability
The probability \(P\) that all five numbers shown will be different is given by the ratio of the number of favorable outcomes to the total sample space:
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Sample Space}} = \frac{720}{7776} \]

### Step 6: Simplify the Probability
To simplify \(\frac{720}{7776}\):
\[ P = \frac{720 \div 720}{7776 \div 720} = \frac{1}{10.8} = \frac{5}{54} \]

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