Three dice are rolled. The probability that different numbers will appear on them is

Options

A. `2//3`
B. `4//9`
C. `5//9`
D. `2//9`.

Answer

To find the probability that different numbers will appear when three dice are rolled, we can follow these steps:

### Step 1: Calculate the Total Number of Outcomes
When rolling a single die, there are 6 possible outcomes (1, 2, 3, 4, 5, 6). When rolling three dice, the total number of outcomes is given by:
\[ \text{Total Outcomes} = 6 \times 6 \times 6 = 6^3 = 216 \]

### Step 2: Calculate the Favorable Outcomes
To find the number of favorable outcomes where all three dice show different numbers, we can select the numbers for each die as follows:

1. **Choose a number for the first die**: There are 6 options (1 to 6).
2. **Choose a number for the second die**: Since it must be different from the first, there are 5 options left.
3. **Choose a number for the third die**: It must be different from the first two, leaving us with 4 options.

Thus, the number of favorable outcomes is:
\[ \text{Favorable Outcomes} = 6 \times 5 \times 4 = 120 \]

### Step 3: Calculate the Probability
The probability \( P \) that all three dice show different numbers is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{120}{216} \]

### Step 4: Simplify the Probability
To simplify \( \frac{120}{216} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 24:
\[ P = \frac{120 \div 24}{216 \div 24} = \frac{5}{9} \]

### Final Answer
Thus, the probability that different numbers will appear on the three dice is:
\[ \frac{5}{9} \]

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