The expression \( 7P5 \) represents the number of permutations of 7 items taken 5 at a time.

The formula for permutations is given by:

\[
nP r = \frac{n!}{(n - r)!}
\]

Where:
- \( n \) is the total number of items,
- \( r \) is the number of items to choose.

For \( 7P5 \):

\[
7P5 = \frac{7!}{(7 - 5)!} = \frac{7!}{2!}
\]

Now, let's calculate it step by step.

### Step 1: Calculate the factorial of 7 (\( 7! \)):

\[
7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040
\]

### Step 2: Calculate the factorial of 2 (\( 2! \)):

\[
2! = 2 \times 1 = 2
\]

### Step 3: Divide \( 7! \) by \( 2! \):

\[
7P5 = \frac{5040}{2} = 2520
\]

### Final Answer:

The value of \( 7P5 \) is **2520**.

Would you like more details or have any questions?

Here are some related questions:
1. What is the difference between permutations and combinations?
2. How do you calculate combinations \( nC r \)?
3. Can you explain the factorial function and its applications?
4. What are real-life examples of permutations?
5. How do you calculate permutations with repetition allowed?

**Tip:** Permutations are used when the order of selection matters, such as in arranging objects in a specific sequence.

Understand how to calculate permutations using the formula nPr = n! / (n - r)!. Step-by-step solution for 7P5 with detailed factorial calculations. Suitable for high school students.

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