Let `A` be a set containing `10` distinct elements. Then the total number of distinct functions from `A` to `A` is:

Options

A. (a) `10!`
B. (b) `10^(10)`
C. (c) `2^(10)`
D. (d) `2^(10)-1`

Explanation

To find the total number of distinct functions from a set \( A \) containing \( 10 \) distinct elements to itself, we can follow these steps:

### Step 1: Understand the problem
We need to determine how many distinct functions can be formed where each function maps elements from set \( A \) to elements in set \( A \).

### Step 2: Define the set
Let \( A = \{ a_1, a_2, a_3, \ldots, a_{10} \} \) be our set containing \( 10 \) distinct elements.

### Step 3: Determine the mapping
For each element in set \( A \), we need to assign an image (output) in set \( A \). Since there are \( 10 \) elements in \( A \), each element can map to any of the \( 10 \) elements in \( A \).

### Step 4: Calculate the number of choices
For each of the \( 10 \) elements in \( A \), we have \( 10 \) choices for its image. Thus, the number of choices for each element is:
- For \( a_1 \): \( 10 \) choices
- For \( a_2 \): \( 10 \) choices
- For \( a_3 \): \( 10 \) choices
- ...
- For \( a_{10} \): \( 10 \) choices

### Step 5: Use the multiplication principle
Since the choices are independent, we can multiply the number of choices for each element:
\[ \text{Total number of functions} = 10 \times 10 \times 10 \times \ldots \text{(10 times)} = 10^{10} \]

### Step 6: Conclusion
Thus, the total number of distinct functions from set \( A \) to set \( A \) is \( 10^{10} \).

### Final Answer
The correct option is \( \text{B: } 10^{10} \).

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