The expression \( 10P7 \) refers to a **permutation** of 10 objects taken 7 at a time. The formula for permutations is:

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

For \( 10P7 \), we have \( n = 10 \) and \( r = 7 \). Applying the formula:

\[
10P7 = \frac{10!}{(10 - 7)!} = \frac{10!}{3!}
\]

Now, we calculate \( 10! \) and \( 3! \):

\[
10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800
\]
\[
3! = 3 \times 2 \times 1 = 6
\]

Finally, we compute the permutation:

\[
10P7 = \frac{3,628,800}{6} = 604,800
\]

So, the value of \( 10P7 \) is **604,800**.

Would you like further details on permutations or have any questions?

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Here are some related questions to consider:
1. What is the difference between permutations and combinations?
2. How would you compute \( 10C7 \) (combinations instead of permutations)?
3. What is the meaning of \( n! \) (n factorial) and how is it used in other areas of mathematics?
4. How do you calculate permutations when some objects are indistinguishable?
5. What real-world problems involve the use of permutations?

**Tip:** Permutations are used when the order of selection matters, while combinations are used when the order does not matter.

Understand how to evaluate the permutation 10P7, using the permutation formula nPr = n! / (n - r)!. This detailed step-by-step solution includes the calculation of factorials, showing how 10! is divided by 3!, resulting in 604,800. Suitable for students in Grades 9-11.

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