Let's break down the two expressions you provided: \( 10C \) and \( 8P5 \).

### 1. Combination: \( 10C \)
This notation is incomplete. It likely refers to the combination formula, denoted as \( \binom{n}{r} \), where \( n \) is the total number of objects, and \( r \) is the number of objects selected. The complete form should be \( \binom{10}{r} \), representing the number of ways to choose \( r \) objects from 10 objects.

The formula for combinations is:

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

Could you clarify what value of \( r \) you are using? For now, I’ll move on to the next part.

### 2. Permutation: \( 8P5 \)
This represents the number of ways to arrange 5 objects selected from a set of 8, and it is calculated using the permutation formula:

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

For \( 8P5 \):

\[
P(8, 5) = \frac{8!}{(8-5)!} = \frac{8!}{3!}
\]

Now let's compute \( 8P5 \).

\[
8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320
\]
\[
3! = 3 \times 2 \times 1 = 6
\]
\[
P(8, 5) = \frac{40320}{6} = 6720
\]

So, \( 8P5 = 6720 \).

Would you like to provide a value for \( r \) in \( 10C \), or should I move forward?

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### Expanding your learning:
1. How are combinations different from permutations?
2. What are some real-world examples where combinations are used?
3. Can you explain why permutations care about order while combinations do not?
4. What happens when \( r = 0 \) in \( nCr \)?
5. How can you generalize the formula for calculating \( nPr \) to different scenarios?

### Tip:
Always remember: **Permutations** deal with **order**; **combinations** do not!

Explore the concepts of combinations and permutations with this detailed explanation of 10C and 8P5. Learn the formulas and methods to calculate permutations and combinations, including examples and step-by-step solutions. Suitable for high school students in Grades 9-12.

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