The arithmaeic mean of the nine numbers in the given set `{9,99,999,…..999999999}` is a 9 digit number N, all whose digits are distinct. Then which digit does not appear in number

Answer

To solve the problem, we need to find the arithmetic mean of the numbers in the set `{9, 99, 999, …, 999999999}` and determine which digit does not appear in the resulting 9-digit number.

### Step-by-Step Solution:

1. **Identify the Numbers in the Set:**
The numbers in the set are:
- 9 (1 digit)
- 99 (2 digits)
- 999 (3 digits)
- 9999 (4 digits)
- 99999 (5 digits)
- 999999 (6 digits)
- 9999999 (7 digits)
- 99999999 (8 digits)
- 999999999 (9 digits)

2. **Calculate the Sum of the Numbers:**
We can express each number in the set as:
- \( 9 = 9 \times 1 \)
- \( 99 = 9 \times 11 \)
- \( 999 = 9 \times 111 \)
- \( 9999 = 9 \times 1111 \)
- \( 99999 = 9 \times 11111 \)
- \( 999999 = 9 \times 111111 \)
- \( 9999999 = 9 \times 1111111 \)
- \( 99999999 = 9 \times 11111111 \)
- \( 999999999 = 9 \times 111111111 \)

Therefore, the sum can be factored as:
\[ \text{Sum} = 9 \times (1 + 11 + 111 + 1111 + 11111 + 111111 + 1111111 + 11111111 + 111111111) \]

3. **Calculate the Sum of the Series:**
The series \( 1 + 11 + 111 + 1111 + \ldots + 111111111 \) can be simplified. Each term can be expressed as:
\[ 1 + 11 + 111 + \ldots + 111111111 = 1 + 10 + 100 + \ldots + 10^8 \]
This is a geometric series with the first term \( a = 1 \) and common ratio \( r = 10 \), having 9 terms.

The sum of a geometric series is given by:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Substituting the values:
\[ S_9 = 1 \frac{10^9 - 1}{10 - 1} = \frac{10^9 - 1}{9} \]

4. **Calculate the Total Sum:**
Now substituting back into the sum:
\[ \text{Sum} = 9 \times \frac{10^9 - 1}{9} = 10^9 - 1 \]

5. **Calculate the Arithmetic Mean:**
The arithmetic mean \( N \) is given by:
\[ N = \frac{\text{Sum}}{9} = \frac{10^9 - 1}{9} \]
This simplifies to:
\[ N = 111111111 \]
This number consists of 9 digits, all of which are '1'.

6. **Identify Distinct Digits:**
The number \( N = 111111111 \) has only one distinct digit, which is '1'.

7. **Determine the Missing Digit:**
The digits from 0 to 9 are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Since '1' is present in \( N \) and all other digits (2 to 9) are missing, the only digit that does not appear in \( N \) is '0'.

### Conclusion:
The digit that does not appear in the number \( N \) is **0**.