A person throws four standard six sided distinguishable dice. The probability that the product of four numbers on the upper faces is 144 is p then 81p is equal to ……..

Answer

To solve the problem, we need to find the probability \( p \) that the product of the numbers on the upper faces of four thrown dice equals 144. Then we will calculate \( 81p \).

### Step-by-Step Solution:

1. **Factorize 144**:
First, we need to factor 144 to understand which combinations of numbers can yield this product.
\[ 144 = 12 \times 12 = (3 \times 2 \times 2) \times (3 \times 2 \times 2) = 3^2 \times 2^4 \]
This means we need to find combinations of the numbers 1 to 6 (the faces of the dice) that multiply to 144.

2. **Identify Possible Combinations**:
We need to find groups of four numbers (from the set {1, 2, 3, 4, 5, 6}) that multiply to 144. The valid combinations are:
- \( (6, 6, 4, 1) \)
- \( (6, 4, 3, 2) \)
- \( (4, 4, 3, 3) \)
- \( (2, 2, 6, 6) \)

3. **Count Arrangements for Each Combination**:
- For \( (6, 6, 4, 1) \):
\[ \text{Arrangements} = \frac{4!}{2!} = 12 \]
- For \( (6, 4, 3, 2) \):
\[ \text{Arrangements} = 4! = 24 \]
- For \( (4, 4, 3, 3) \):
\[ \text{Arrangements} = \frac{4!}{2! \times 2!} = 6 \]
- For \( (2, 2, 6, 6) \):
\[ \text{Arrangements} = \frac{4!}{2! \times 2!} = 6 \]

4. **Total Favorable Outcomes**:
Now, we sum the arrangements:
\[ \text{Total Favorable Outcomes} = 12 + 24 + 6 + 6 = 48 \]

5. **Calculate Total Outcomes**:
The total number of outcomes when throwing four dice is:
\[ 6^4 = 1296 \]

6. **Calculate Probability \( p \)**:
The probability \( p \) that the product of the numbers on the upper faces is 144 is:
\[ p = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{48}{1296} = \frac{1}{27} \]

7. **Calculate \( 81p \)**:
Now we calculate \( 81p \):
\[ 81p = 81 \times \frac{1}{27} = 3 \]

### Final Answer:
Thus, \( 81p = 3 \).