The number of bijective functions from set A to itself when A contains 106 elements is

Options

A. 106
B. `(106)^(2)`
C. 106!
D. `2^(106)`

Answer

To find the number of bijective functions from set A to itself when A contains 106 elements, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Bijective Functions**: A function is bijective if it is both injective (one-to-one) and surjective (onto). This means that every element in set A must be mapped to a unique element in set A, and every element in set A must be the image of some element in set A.

2. **Identifying Set A**: Let’s denote set A as containing elements {1, 2, 3, ..., 106}. This means that the size of set A, denoted as |A|, is 106.

3. **Counting Bijective Functions**: The number of bijective functions from a set of n elements to itself is given by the number of permutations of n elements. For a set of size n, the number of permutations is n factorial (n!).

4. **Calculating 106 Factorial**: Since our set A has 106 elements, the number of bijective functions from A to itself is:
\[ |A|! = 106! \]

5. **Conclusion**: Therefore, the number of bijective functions from set A to itself when A contains 106 elements is \( 106! \).

### Final Answer:
The number of bijective functions from set A to itself when A contains 106 elements is \( 106! \).

---