An oridinary cube has four faces, one face marked 2 another marked 3, Then the probability of obtaining a total of exactly 12 in five throws is

Options

A. `(5)/(1296)`
B. `(5)/(1944)`
C. `(5)/(2592)`
D. none of these

Answer

To solve the problem of finding the probability of obtaining a total of exactly 12 in five throws of a cube with specific markings, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Cube Configuration**:
- The cube has 6 faces: 4 faces are blank (0), 1 face is marked with 2, and 1 face is marked with 3.
- Therefore, the possible outcomes when rolling the cube are: 0, 0, 0, 0, 2, 3.

2. **Determine the Total Outcomes**:
- Each throw of the cube has 6 possible outcomes.
- Since we are throwing the cube 5 times, the total number of possible outcomes (sample space) is:
\[ \text{Total Outcomes} = 6^5 = 7776 \]

3. **Identify Favorable Outcomes**:
- We need to find combinations of the numbers rolled that sum to exactly 12 in 5 throws.
- The possible combinations to achieve a total of 12 can be:
- **Case 1**: 4 faces showing 3 and 1 face showing 0.
- This gives us: \(3 + 3 + 3 + 3 + 0 = 12\).
- The number of ways to arrange this is given by choosing 1 blank face from 5 throws:
\[ \text{Ways} = \binom{5}{1} = 5 \]
- **Case 2**: 2 faces showing 3 and 3 faces showing 2.
- This gives us: \(3 + 3 + 2 + 2 + 2 = 12\).
- The number of ways to arrange this is given by choosing 2 faces showing 3 from 5 throws:
\[ \text{Ways} = \binom{5}{2} = 10 \]

4. **Calculate Total Favorable Outcomes**:
- Adding the favorable outcomes from both cases:
\[ \text{Total Favorable Outcomes} = 5 + 10 = 15 \]

5. **Calculate the Probability**:
- The probability of obtaining a total of exactly 12 in 5 throws is given by the ratio of favorable outcomes to total outcomes:
\[ P(\text{Total} = 12) = \frac{\text{Total Favorable Outcomes}}{\text{Total Outcomes}} = \frac{15}{7776} \]
- Simplifying this fraction:
\[ P(\text{Total} = 12) = \frac{15}{7776} = \frac{5}{2592} \]

### Final Answer:
The probability of obtaining a total of exactly 12 in five throws is:
\[ \frac{5}{2592} \]