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 on three rolled dice, we can follow these steps:

### Step 1: Determine the total number of outcomes when rolling three dice.
Each die has 6 faces, so when rolling three dice, the total number of outcomes is calculated as follows:
\[ \text{Total outcomes} = 6 \times 6 \times 6 = 216 \]

### Step 2: Calculate the number of favorable outcomes where all three dice show different numbers.
To find the number of ways to roll three dice such that all show different numbers, we can select numbers for each die as follows:

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

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

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

### 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:
\[ \frac{120 \div 24}{216 \div 24} = \frac{5}{9} \]

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

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