A 4 digit number is randomly picked from all the 4 digit numbers, then the probability that the product of its digit is divisible by 3 is :

Options

A. `(107)/(125)`
B. `(109)/(125)`
C. `(111)/(125)`
D. None of these

Answer

To solve the problem of finding the probability that the product of the digits of a randomly picked 4-digit number is divisible by 3, we will follow these steps:

### Step 1: Determine the total number of 4-digit numbers
A 4-digit number cannot start with 0. Therefore, for the first digit, we have 9 choices (1 to 9). For the remaining three digits, we can use any digit from 0 to 9, giving us 10 choices for each of those places.

**Calculation:**
- Choices for the first digit: 9 (1-9)
- Choices for the second digit: 10 (0-9)
- Choices for the third digit: 10 (0-9)
- Choices for the fourth digit: 10 (0-9)

Total number of 4-digit numbers = \( 9 \times 10^3 = 9000 \).

### Step 2: Identify the digits that make the product divisible by 3
A product of digits is divisible by 3 if at least one of the digits is a multiple of 3. The digits that are multiples of 3 from 0 to 9 are 0, 3, 6, and 9.

**Multiples of 3:** 0, 3, 6, 9

### Step 3: Calculate the probability of the product not being divisible by 3
To find the probability that the product is divisible by 3, we will first calculate the probability that it is NOT divisible by 3. This occurs when none of the digits are 3, 6, or 9.

The digits that are NOT multiples of 3 are: 1, 2, 4, 5, 7, 8 (total of 6 digits).

**Choices:**
- For the first digit (cannot be 0 and must not be 3, 6, or 9): 6 choices (1, 2, 4, 5, 7, 8)
- For the second digit: 6 choices (1, 2, 4, 5, 7, 8)
- For the third digit: 6 choices (1, 2, 4, 5, 7, 8)
- For the fourth digit: 6 choices (1, 2, 4, 5, 7, 8)

Total number of 4-digit numbers where the product is NOT divisible by 3 = \( 6 \times 6 \times 6 \times 6 = 6^4 = 1296 \).

### Step 4: Calculate the probability
Now we can find the probability that the product of the digits is divisible by 3 using the complement rule.

**Probability that the product is NOT divisible by 3:**
\[ P(A') = \frac{1296}{9000} \]

**Probability that the product is divisible by 3:**
\[ P(A) = 1 - P(A') = 1 - \frac{1296}{9000} = \frac{9000 - 1296}{9000} = \frac{7704}{9000} \]

### Step 5: Simplify the probability
To simplify \( \frac{7704}{9000} \):
- Divide both the numerator and the denominator by 36 (the GCD):
\[ \frac{7704 \div 36}{9000 \div 36} = \frac{107}{125} \]

### Final Answer
The probability that the product of the digits of a randomly picked 4-digit number is divisible by 3 is:
\[ \frac{107}{125} \]