The number of one-to-one functions from an 7-element set to itself isa. 64.00b. 128.00c. 7!d. None of the option is correcte. 2187.00

To solve the problem of finding the number of one-to-one functions from a 7-element set to itself, we can follow these steps:

1. Break Down the Problem

We need to determine the number of one-to-one (or injective) functions from a set with 7 elements to itself. A one-to-one function means that each element in the domain maps to a unique element in the codomain.

2. Relevant Concepts

For a set with $n$ elements, the number of one-to-one functions from the set to itself is given by the formula $n!$ (n factorial). For our case, $n = 7$.

3. Analysis and Detail

Let's calculate $7!$:

$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$

Calculating step by step:

• $7 \times 6 = 42$
• $42 \times 5 = 210$
• $210 \times 4 = 840$
• $840 \times 3 = 2520$
• $2520 \times 2 = 5040$
• $5040 \times 1 = 5040$

So, $7! = 5040$.

4. Verify and Summarize

Now we need to verify our options against the calculated result of $7! = 5040$.

None of the options a (64.00), b (128.00), c (7!), or d (None of the option is correct; 2187.00) mention 5040.

Final Answer

The correct answer is $\mathbf{5040}$ (which corresponds to 7!). Therefore, the option would be d. None of the option is correct.