The number of `4` digit natural numbers such that the product of their digits is `12` is

Options

A. `24`
B. `36`
C. `42`
D. `48`

Explanation

To find the number of 4-digit natural numbers such that the product of their digits is 12, we can follow these steps:

### Step 1: Factorization of 12
First, we need to find the factors of 12. The prime factorization of 12 is:
\[ 12 = 2^2 \times 3^1 \times 1^1 \]
This means we can use the digits 2, 2, 3, and 1 to form our 4-digit number.

### Step 2: Arranging the Digits
We can arrange the digits 2, 2, 3, and 1. Since the digit '2' is repeated, we will use the formula for permutations of multiset:
\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots} \]
where \( n \) is the total number of items to arrange, and \( p_1, p_2, \ldots \) are the counts of each repeated item.

For the digits 2, 2, 3, and 1:
- Total digits \( n = 4 \)
- Repeated digit '2' occurs \( 2 \) times.

So the number of arrangements is:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]

### Step 3: Other Combinations
Next, we need to consider other combinations of digits that multiply to 12. The possible combinations of digits that can give us a product of 12 are:

1. **Combination 1**: \( 1, 1, 2, 6 \)
- Here, '1' is repeated twice.
- Arrangements:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]

2. **Combination 2**: \( 1, 1, 3, 4 \)
- Again, '1' is repeated twice.
- Arrangements:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]

### Step 4: Total Count
Now we add the arrangements from all combinations:
- From \( 2, 2, 3, 1 \): 12 arrangements
- From \( 1, 1, 2, 6 \): 12 arrangements
- From \( 1, 1, 3, 4 \): 12 arrangements

Total arrangements:
\[ 12 + 12 + 12 = 36 \]

### Conclusion
The total number of 4-digit natural numbers such that the product of their digits is 12 is:
\[ \boxed{36} \]