If A={1,2,3,4} and f : A->A, then total number of invertible functions,'f',such that `f(2)!=2`,`f(4)!=4`,`f(1)=1` is equal to:

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the total number of invertible functions \( f: A \to A \) under the given conditions:

1. \( f(1) = 1 \)
2. \( f(2) \neq 2 \)
3. \( f(4) \neq 4 \)

Given the set \( A = \{1, 2, 3, 4\} \), we can analyze the conditions step by step.

### Step 1: Fix \( f(1) \)
Since \( f(1) = 1 \), we have already determined the image of 1. The function \( f \) is now defined partially as:
- \( f(1) = 1 \)

### Step 2: Determine possible values for \( f(2) \)
The condition \( f(2) \neq 2 \) means that \( f(2) \) can either be 3 or 4. We will consider both cases.

### Case 1: \( f(2) = 3 \)
- Now we have:
- \( f(1) = 1 \)
- \( f(2) = 3 \)

Since \( f \) is a function from \( A \) to \( A \), the remaining values for \( f(3) \) and \( f(4) \) can only be 2 or 4. However, we also have the condition \( f(4) \neq 4 \), which means:
- \( f(4) = 2 \)
- Thus, \( f(3) \) must be 4.

So, in this case, we have:
- \( f(1) = 1 \)
- \( f(2) = 3 \)
- \( f(3) = 4 \)
- \( f(4) = 2 \)

This gives us one valid function.

### Case 2: \( f(2) = 4 \)
- Now we have:
- \( f(1) = 1 \)
- \( f(2) = 4 \)

In this case, \( f(3) \) can only be 2 or 3. However, since \( f(4) \neq 4 \), we have:
- If \( f(3) = 2 \), then \( f(4) \) must be 3.
- If \( f(3) = 3 \), then \( f(4) \) must be 2.

So we have two subcases:
1. \( f(3) = 2 \) and \( f(4) = 3 \)
2. \( f(3) = 3 \) and \( f(4) = 2 \)

Thus, we have two valid functions from this case.

### Summary of Valid Functions
1. From Case 1:
- \( f(1) = 1, f(2) = 3, f(3) = 4, f(4) = 2 \)
2. From Case 2:
- \( f(1) = 1, f(2) = 4, f(3) = 2, f(4) = 3 \)
- \( f(1) = 1, f(2) = 4, f(3) = 3, f(4) = 2 \)

### Total Count of Invertible Functions
Adding all valid functions together, we find:
- 1 function from Case 1
- 2 functions from Case 2

Thus, the total number of invertible functions \( f \) is \( 1 + 2 = 3 \).

### Final Answer
The total number of invertible functions \( f \) such that \( f(2) \neq 2 \), \( f(4) \neq 4 \), and \( f(1) = 1 \) is **3**.