Two 8-faced dice (numbered from 1 to 8) are tossed. The probability that the product of two counts is a square number , is

Options

A. `(1)/(8)`
B. `(12)/(6^2)`
C. `(3)/(16)`
D. `(12)/(8^2)`.

Answer

To solve the problem of finding the probability that the product of two counts from tossing two 8-faced dice is a square number, we will follow these steps:

### Step 1: Determine the Sample Space
When two 8-faced dice are tossed, each die has 8 faces. Therefore, the total number of outcomes when rolling two dice is:
\[ 8 \times 8 = 64 \]
**Hint:** Remember that the total outcomes can be calculated by multiplying the number of outcomes for each die.

### Step 2: Identify Perfect Square Products
Next, we need to identify the perfect squares that can result from the product of the numbers on the two dice. The perfect squares from 1 to 64 are:
\[ 1, 4, 9, 16, 25, 36, 49, 64 \]
**Hint:** A perfect square is a number that can be expressed as the square of an integer.

### Step 3: Count Favorable Outcomes for Each Perfect Square
Now, we will find the pairs of numbers (from the two dice) that yield each of these perfect squares:

- **For 1:** (1, 1) → 1 way
- **For 4:** (1, 4), (2, 2), (4, 1) → 3 ways
- **For 9:** (3, 3) → 1 way
- **For 16:** (2, 8), (4, 4), (8, 2) → 3 ways
- **For 25:** (5, 5) → 1 way (but 5 is not possible on an 8-faced die, so 0 ways)
- **For 36:** (6, 6) → 1 way (but 6 is not possible on an 8-faced die, so 0 ways)
- **For 49:** (7, 7) → 1 way (but 7 is not possible on an 8-faced die, so 0 ways)
- **For 64:** (8, 8) → 1 way

Now, we sum the valid outcomes:
- For 1: 1 way
- For 4: 3 ways
- For 9: 1 way
- For 16: 3 ways
- For 25: 0 ways
- For 36: 0 ways
- For 49: 0 ways
- For 64: 1 way

Total favorable outcomes:
\[ 1 + 3 + 1 + 3 + 0 + 0 + 0 + 1 = 9 \]
**Hint:** Make sure to check each perfect square and count only valid pairs.

### Step 4: Calculate the Probability
The probability \( P \) that the product of the two counts is a perfect square is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{9}{64} \]
**Hint:** Probability is calculated as the number of successful outcomes divided by the total number of possible outcomes.

### Step 5: Simplify the Probability
In this case, \( \frac{9}{64} \) is already in its simplest form.
**Hint:** Always check if the fraction can be simplified further.

### Final Answer
The probability that the product of the two counts is a square number is:
\[ \frac{9}{64} \]