Digits 1, 2, 3, ….. 9 are written in a random order to form a 9 digit number. If the number happens to be divisible by 4. Then probability that it will be divisible by 36 is given by

Options

A. `2/9`
B. `3/4`
C. `7/9`
D. 1

Answer

To solve the problem, we need to find the probability that a 9-digit number formed by the digits 1 to 9 is divisible by 36, given that it is divisible by 4.

### Step-by-Step Solution:

1. **Understanding Divisibility by 36**:
A number is divisible by 36 if it is divisible by both 4 and 9.

2. **Divisibility by 4**:
For a number to be divisible by 4, its last two digits must form a number that is divisible by 4. Since we are using the digits 1 to 9, the possible pairs of last two digits that are even and divisible by 4 are:
- 12, 16, 24, 32, 36, 52, 56, 72, 76, 92, 84, 28, 48, 68 (total of 16 valid pairs).

3. **Counting Arrangements**:
After selecting a valid pair for the last two digits, we have 7 remaining digits to arrange. The number of arrangements of these 7 digits is \(7!\).

4. **Total Arrangements**:
The total number of arrangements of the digits 1 to 9 is \(9!\).

5. **Calculating Probability of Divisibility by 4**:
The probability \(P(E_1)\) that a randomly formed number is divisible by 4 is given by:
\[ P(E_1) = \frac{16 \times 7!}{9!} \]
Simplifying this, we get:
\[ P(E_1) = \frac{16}{9 \times 8} = \frac{2}{9} \]

6. **Divisibility by 9**:
Since the sum of the digits from 1 to 9 is 45, which is divisible by 9, any arrangement of these digits will also be divisible by 9.

7. **Conclusion on Divisibility by 36**:
Therefore, a number formed by the digits 1 to 9 is divisible by 36 if it is divisible by 4 (since it is always divisible by 9). Hence, the probability \(P(E_2)\) that a number is divisible by 36 given that it is divisible by 4 is:
\[ P(E_2 | E_1) = 1 \]

8. **Final Probability Calculation**:
The final probability that a number is divisible by 36 given that it is divisible by 4 is:
\[ P(E_2 | E_1) = \frac{P(E_2)}{P(E_1)} = \frac{1}{\frac{2}{9}} = \frac{9}{2} \]

### Final Answer:
The probability that the number will be divisible by 36 given that it is divisible by 4 is \(1\).