A dice has four blank faces and two faces marked 3. The change of getting a total of 12 in 5 throws is

Options

A. `""^(5)C_(4)((1)/(3))(4)((2)/(3))`
B. `""^(5)C_(4)((1)/(3))((2)/(3))^(4)`
C. `""^(5)C_(4)((1)/(6))^(5)`
D. `""^(5)C_(4)((1)/(6))^(4)((5)/(6))`

Explanation

To solve the problem of finding the probability of getting a total of 12 in 5 throws of a dice with 4 blank faces and 2 faces marked with 3, we can follow these steps:

### Step 1: Understand the Dice Configuration
The dice has:
- 4 blank faces (which contribute 0 to the total)
- 2 faces marked with 3

### Step 2: Determine the Probability of Success and Failure
- Probability of rolling a 3 (success) = Number of faces marked with 3 / Total faces = 2/6 = 1/3
- Probability of rolling a blank face (failure) = Number of blank faces / Total faces = 4/6 = 2/3

### Step 3: Set Up the Binomial Distribution
We need to find the number of ways to achieve a total of 12 in 5 throws. To get a total of 12, we can only achieve this by rolling a 3 four times and a blank face once. This is because:
- 4 rolls of 3 contribute 4 * 3 = 12
- 1 roll of blank contributes 0

### Step 4: Calculate the Number of Successful Outcomes
The number of successful outcomes (rolling 3 four times and a blank face once) can be calculated using the binomial coefficient:
- Number of ways to choose 4 successful rolls out of 5 = \( \binom{5}{4} \)

### Step 5: Calculate the Probability
Now we can calculate the probability of this specific outcome:
- Probability of getting 3 four times = \( \left(\frac{1}{3}\right)^4 \)
- Probability of getting a blank face once = \( \left(\frac{2}{3}\right)^1 \)

Thus, the total probability \( P \) is given by:
\[ P = \binom{5}{4} \left(\frac{1}{3}\right)^4 \left(\frac{2}{3}\right)^1 \]

### Step 6: Substitute Values and Calculate
Now we can substitute the values:
- \( \binom{5}{4} = 5 \)
- \( \left(\frac{1}{3}\right)^4 = \frac{1}{81} \)
- \( \left(\frac{2}{3}\right)^1 = \frac{2}{3} \)

Putting it all together:
\[ P = 5 \times \frac{1}{81} \times \frac{2}{3} = 5 \times \frac{2}{243} = \frac{10}{243} \]

### Final Answer
The probability of getting a total of 12 in 5 throws is \( \frac{10}{243} \).

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