The number of increasing function from `f : AtoB` where `A in {a_(1),a_(2),a_(3),a_(4),a_(5),a_(6)}`, `B in {1,2,3,….,9}` such that `a_(i+1) gt a_(i) AA I in N` and `a_(i) ne i` is

Options

A. `30`
B. `28`
C. `24`
D. `42`

Explanation

To find the number of increasing functions \( f: A \to B \) where \( A = \{ a_1, a_2, a_3, a_4, a_5, a_6 \} \) and \( B = \{ 1, 2, 3, \ldots, 9 \} \), with the conditions that \( a_{i+1} > a_i \) for all \( i \) and \( a_i \neq i \), we can follow these steps:

### Step 1: Understanding the Problem
We need to create an increasing sequence of 6 elements from the set \( B \) which contains numbers from 1 to 9. The increasing condition implies that we must choose 6 distinct numbers from \( B \) such that they are in increasing order.

### Step 2: Exclude Invalid Values
The condition \( a_i \neq i \) means that:
- \( a_1 \neq 1 \)
- \( a_2 \neq 2 \)
- \( a_3 \neq 3 \)
- \( a_4 \neq 4 \)
- \( a_5 \neq 5 \)
- \( a_6 \neq 6 \)

This means that we cannot choose the first 6 numbers directly as they would violate the condition.

### Step 3: Choose Values from B
Since we are selecting 6 values from \( B \) (which has 9 elements), we can only select from the numbers 2 to 9 for \( a_1 \) to \( a_6 \).

### Step 4: Calculate the Valid Selections
We can think of the valid selections as follows:
- If we choose \( a_1 \) from \( \{2, 3, 4, 5, 6, 7, 8, 9\} \), we have 8 options.
- If we choose \( a_2 \) from \( \{3, 4, 5, 6, 7, 8, 9\} \), we have 7 options.
- Continuing this way, we can see that we need to ensure that we skip the indices that would violate \( a_i \neq i \).

### Step 5: Case Analysis
We can analyze the cases based on the minimum value of \( a_1 \):
1. **Case 1**: If \( a_1 = 2 \), then we can choose from \( \{3, 4, 5, 6, 7, 8, 9\} \) (7 options left).
2. **Case 2**: If \( a_1 = 3 \), then we can choose from \( \{4, 5, 6, 7, 8, 9\} \) (6 options left).
3. **Case 3**: If \( a_1 = 4 \), then we can choose from \( \{5, 6, 7, 8, 9\} \) (5 options left).
4. **Case 4**: If \( a_1 = 5 \), then we can choose from \( \{6, 7, 8, 9\} \) (4 options left).
5. **Case 5**: If \( a_1 = 6 \), then we can choose from \( \{7, 8, 9\} \) (3 options left).
6. **Case 6**: If \( a_1 = 7 \), then we can choose from \( \{8, 9\} \) (2 options left).
7. **Case 7**: If \( a_1 = 8 \), then we can only choose \( 9 \) (1 option left).

### Step 6: Calculate the Total
Now we can calculate the total number of valid increasing functions by summing the valid selections from each case:
- Case 1: Choose 5 from 7 options: \( \binom{7}{5} = 21 \)
- Case 2: Choose 5 from 6 options: \( \binom{6}{5} = 6 \)
- Case 3: Choose 5 from 5 options: \( \binom{5}{5} = 1 \)

Adding these gives:
\[ 21 + 6 + 1 = 28 \]

### Final Answer
Thus, the total number of increasing functions \( f: A \to B \) that satisfy the given conditions is **28**.