If 2 dice are thrown, then probability that product of the numbers on them is 12 (or that they are factors of 12 ) is

Options

A. `1/6`
B. `1/4`
C. `1/9`
D. `2/7`

Answer

To solve the problem, we need to find the probability that the product of the numbers on two thrown dice is 12. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Total Outcomes
When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing two dice is:
\[ \text{Total Outcomes} = 6 \times 6 = 36 \]

### Step 2: Identify the Favorable Outcomes
Next, we need to find the pairs of numbers (from the two dice) whose product equals 12. We can list the pairs systematically:

- The factors of 12 that can appear on the dice are:
- \(1 \times 12\) (not possible since 12 is not on a die)
- \(2 \times 6\) (possible)
- \(3 \times 4\) (possible)
- \(4 \times 3\) (possible)
- \(6 \times 2\) (possible)

From the above, the valid pairs (considering the order of the dice) that yield a product of 12 are:
1. (2, 6)
2. (6, 2)
3. (3, 4)
4. (4, 3)

Thus, the favorable outcomes are:
- (2, 6)
- (6, 2)
- (3, 4)
- (4, 3)

This gives us a total of 4 favorable outcomes.

### Step 3: Calculate the Probability
The probability \(P\) of an event is given by the formula:
\[ P(\text{event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} \]
Substituting the values we found:
\[ P(\text{product is 12}) = \frac{4}{36} = \frac{1}{9} \]

### Final Answer
The probability that the product of the numbers on the two dice is 12 is:
\[ \frac{1}{9} \]

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