Math Snap

STEP 1

What is this asking?
We need to calculate the fraction of the number of permutations of 10 items taken 8 at a time divided by the number of combinations of 10 items taken 4 at a time.
Watch out!
Don't mix up permutations and combinations!
Remember, order matters for permutations, but not for combinations.

STEP 2

1. Calculate the Permutation
2. Calculate the Combination
3. Calculate the Fraction

STEP 3

Let's start with the permutation!
We're looking at $_{10}P_8$, which means the number of ways to arrange 8 items out of a set of 10, where the order does matter.

STEP 4

The formula for permutations is $_{n}P_r = \frac{n!}{(n-r)!}$.
In our case, $n = \textbf{10}$ and $r = \textbf{8}$, so we have $_{10}P_8 = \frac{10!}{(10-8)!}$.

STEP 5

This simplifies to $_{10}P_8 = \frac{10!}{2!} = \frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{2 \cdot 1}$.
We can cancel out the $2 \cdot 1$ in the numerator and denominator by multiplying both by $\frac{1}{2}$, giving us $10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 = \textbf{1,814,400}$.
So, there are 1,814,400 ways to arrange 8 items out of 10.

STEP 6

Now, let's tackle the combination!
We have $_{10}C_4$, which represents the number of ways to choose 4 items out of 10, where the order doesn't matter.

STEP 7

The formula for combinations is $_{n}C_r = \frac{n!}{r!(n-r)!}$.
Here, $n = \textbf{10}$ and $r = \textbf{4}$, so $_{10}C_4 = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!}$.

STEP 8

Let's break this down: $\frac{10!}{4!6!} = \frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(4 \cdot 3 \cdot 2 \cdot 1)(6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$.
We can cancel out the $6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ from both the numerator and denominator by multiplying both by $\frac{1}{6!}$, which simplifies to $\frac{10 \cdot 9 \cdot 8 \cdot 7}{4 \cdot 3 \cdot 2 \cdot 1}$.

STEP 9

We can simplify further: $\frac{10 \cdot 9 \cdot 8 \cdot 7}{4 \cdot 3 \cdot 2 \cdot 1} = \frac{5040}{24} = \textbf{210}$.
There are 210 ways to choose 4 items out of 10.

STEP 10

Finally, we need to calculate the fraction: $\frac{_{10}P_8}{_{10}C_4}$.
We already calculated the permutation $_{10}P_8 = \textbf{1,814,400}$ and the combination $_{10}C_4 = \textbf{210}$.

STEP 11

So, the fraction is $\frac{1,814,400}{210} = \textbf{8,640}$.

The final answer is 8,640.