Seven digits from the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 are written in random order. The probability that this seven digit number is divisible by 9 is :

Options

A. a. `7/9`
B. b. `1/9`
C. c. `2/(9!)`
D. d. `4/9`

Answer

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

### Step 1: Understanding the Divisibility Rule for 9
A number is divisible by 9 if the sum of its digits is divisible by 9.

**Hint:** Remember that the sum of the digits is crucial for checking divisibility by 9.

### Step 2: Calculate the Total Sum of Digits from 1 to 9
The digits from 1 to 9 are: 1, 2, 3, 4, 5, 6, 7, 8, 9.
The sum of these digits is:
\[ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 \]
Since 45 is divisible by 9, any combination of 7 digits must have a sum that is also divisible by 9.

**Hint:** Calculate the total sum of the digits to understand the overall divisibility.

### Step 3: Determine the Sum of the Excluded Digits
To form a seven-digit number, we need to exclude 2 digits from the total of 9 digits. The sum of the excluded digits must also be divisible by 9 for the sum of the remaining digits (which will be the sum of the seven digits) to be divisible by 9.

**Hint:** Focus on the sum of the digits you choose to exclude.

### Step 4: Find Combinations of Excluded Digits
We need to find pairs of digits whose sum is divisible by 9. The possible pairs of digits that can be excluded are:
- Exclude 1 and 8: Sum = 9
- Exclude 2 and 7: Sum = 9
- Exclude 3 and 6: Sum = 9
- Exclude 4 and 5: Sum = 9

Thus, there are 4 valid pairs of digits that can be excluded.

**Hint:** List out pairs of digits and check their sums for divisibility by 9.

### Step 5: Calculate the Total Ways to Choose 7 Digits
The total number of ways to choose 7 digits from 9 is given by the combination formula:
\[ \binom{9}{7} = \binom{9}{2} = \frac{9!}{7! \cdot 2!} = 36 \]

**Hint:** Use the combination formula to find the total selections.

### Step 6: Calculate the Probability
The probability \( P \) that a randomly selected combination of 7 digits is divisible by 9 is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{4}{36} = \frac{1}{9} \]

**Hint:** Simplify the fraction to find the final probability.

### Final Answer
The probability that the seven-digit number is divisible by 9 is \( \frac{1}{9} \).