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!