Three identical dice are thrown together. Find the probability that distinct numbers appear on them.

Options

A. `(4)/(9)`
B. `(5)/(9)`
C. `(5)/(39)`
D. `(1)/(9)`

Explanation

To find the probability that distinct numbers appear on three identical dice thrown together, we can follow these steps:

### Step 1: Calculate the Total Number of Outcomes
When three dice are thrown, each die can show one of six faces (1 to 6). Therefore, the total number of outcomes when throwing three dice is calculated as:
\[ \text{Total Outcomes} = 6 \times 6 \times 6 = 6^3 = 216 \]

### Step 2: Calculate the Number of Favorable Outcomes for Distinct Numbers
To find the number of ways to get three distinct numbers, we first need to choose 3 different numbers from the 6 available numbers on the dice. The number of ways to choose 3 distinct numbers from 6 is given by the combination formula:
\[ \text{Ways to choose 3 distinct numbers} = \binom{6}{3} \]
Calculating this:
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 3: Arranging the Chosen Numbers
Once we have chosen 3 distinct numbers, we can arrange them in different orders. The number of arrangements of 3 distinct numbers is given by:
\[ 3! = 6 \]

### Step 4: Calculate the Total Number of Favorable Outcomes
Now, we can calculate the total number of favorable outcomes where all three dice show distinct numbers:
\[ \text{Favorable Outcomes} = \text{Ways to choose 3 distinct numbers} \times \text{Arrangements of those numbers} = 20 \times 6 = 120 \]

### Step 5: Calculate the Probability
Finally, the probability that all three dice show distinct numbers can be calculated as:
\[ \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{120}{216} \]
To simplify this fraction:
\[ \frac{120}{216} = \frac{5}{9} \]

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

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