Let A denote the set of all 4-digit natural numbers with no digit being 0. Let `B subsetA` consist of all numbers x such that no permutation of the digits of x gives a number that is divisible by 4. Then the probability of drawing a number from B with all even digits is

Options

A. `625/1641`
B. `16/641`
C. `16/1641`
D. `1000/1641`

Answer

To solve the problem step by step, we will first define the sets and then calculate the required probability.

### Step 1: Define the Set A
Let \( A \) be the set of all 4-digit natural numbers with no digit being 0. The digits can be from 1 to 9.

### Step 2: Count the Total Numbers in Set A
The total number of 4-digit numbers can be calculated as follows:
- The first digit can be any digit from 1 to 9 (9 choices).
- The second digit can also be any digit from 1 to 9 (9 choices).
- The third digit can also be any digit from 1 to 9 (9 choices).
- The fourth digit can also be any digit from 1 to 9 (9 choices).

Thus, the total number of 4-digit numbers in set \( A \) is:
\[ |A| = 9 \times 9 \times 9 \times 9 = 9^4 = 6561 \]

### Step 3: Define the Set B
Let \( B \subset A \) be the subset of numbers such that no permutation of the digits of \( x \) gives a number that is divisible by 4. A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

### Step 4: Identify Even Digits
The even digits available (since no digit can be 0) are 2, 4, 6, and 8.

### Step 5: Count the Numbers in Set B with All Even Digits
To find the numbers in \( B \) that consist of all even digits, we can only use the digits 2, 4, 6, and 8.

1. **Case 1**: All four digits are even.
- The first digit can be any of the 4 even digits (2, 4, 6, 8).
- The second digit can also be any of the 4 even digits.
- The third digit can also be any of the 4 even digits.
- The fourth digit can also be any of the 4 even digits.

Thus, the total number of 4-digit numbers with all even digits is:
\[ |B| = 4 \times 4 \times 4 \times 4 = 4^4 = 256 \]

### Step 6: Calculate the Probability
The probability of drawing a number from \( B \) with all even digits is given by the ratio of the size of \( B \) to the size of \( A \):
\[ P = \frac{|B|}{|A|} = \frac{256}{6561} \]

### Step 7: Simplify the Probability
This fraction cannot be simplified further, so we leave it as:
\[ P = \frac{256}{6561} \]

### Final Answer
The probability of drawing a number from \( B \) with all even digits is:
\[ \frac{256}{6561} \]