Let `A={1,2,3,4,5}` and `f:A rarr A` be an into function such that `f(x) nex forall x in A`. Then number of such functions `f` is:

Options

A. (a) `1024`
B. (b) `904`
C. (c) `980`
D. (d) None of these

Answer

To solve the problem of finding the number of into functions \( f: A \to A \) such that \( f(x) \neq x \) for all \( x \in A \), we will follow these steps:

### Step 1: Understand the Problem
We have a set \( A = \{1, 2, 3, 4, 5\} \) and we need to find functions \( f \) that are into (meaning the range is a proper subset of the codomain) and also satisfy the condition \( f(x) \neq x \) for all \( x \in A \).

### Step 2: Identify the Characteristics of the Function
An "into" function means that not all elements in the codomain (which is also \( A \)) are used as images. Since \( f(x) \neq x \), each element in \( A \) cannot map to itself. Thus, for each \( x \) in \( A \), it can only map to one of the other four elements in \( A \).

### Step 3: Count the Possible Mappings
For each of the 5 elements in \( A \):
- Each element can map to any of the 4 other elements (since it cannot map to itself).

Thus, for each of the 5 elements, there are 4 choices. Therefore, the total number of functions \( f \) can be calculated as follows:

\[ \text{Total functions} = 4^5 \]

### Step 4: Calculate the Total
Now we compute \( 4^5 \):

\[ 4^5 = 1024 \]

### Step 5: Conclusion
The total number of such functions \( f \) is \( 1024 \).

Thus, the answer to the question is:

**Option A: 1024**

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