The digits 1,2,3,4,5,6,7,8 and 9 are written in random order to form a nine digit number. The probability that this number is divisible by 4, is

Options

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

Explanation

To solve the problem of finding the probability that a nine-digit number formed by the digits 1 to 9 is divisible by 4, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Divisibility by 4**:
A number is divisible by 4 if the number formed by its last two digits is divisible by 4. Therefore, we need to focus on the last two digits of our nine-digit number.

2. **Identifying Possible Last Two Digits**:
The digits available to form the last two digits are 1, 2, 3, 4, 5, 6, 7, 8, and 9. We need to find all the pairs of these digits that can form a two-digit number divisible by 4.

3. **Finding Valid Combinations**:
The valid two-digit combinations (where the first digit can be any of the digits and the second digit can be any of the remaining digits) that are divisible by 4 are:
- 12
- 16
- 24
- 32
- 36
- 52
- 56
- 72
- 76
- 84
- 92
- 96

We can check each pair to confirm they are divisible by 4. After checking, we find that there are a total of **12 valid combinations**.

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

5. **Calculating Favorable Outcomes**:
For each valid pair of last two digits, the remaining 7 digits can be arranged in any order. Therefore, for each of the 12 valid pairs, there are \(7!\) arrangements of the remaining digits. Thus, the total number of favorable outcomes is:
\[ \text{Favorable Outcomes} = 12 \times 7! \]

6. **Calculating Probability**:
The probability \(P\) that a randomly arranged nine-digit number is divisible by 4 is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{12 \times 7!}{9!} \]
Simplifying this, we know that \(9! = 9 \times 8 \times 7!\), so:
\[ P = \frac{12 \times 7!}{9 \times 8 \times 7!} = \frac{12}{72} = \frac{1}{6} \]

### Final Answer:
The probability that the nine-digit number is divisible by 4 is \(\frac{1}{6}\).