Math / Discrete

QuestionEvaluate the following expression. Express your answer as a fraction or a decimal number rounded to four decimal places. $\frac{{ }_{10} P_{8}}{{ }_{10} C_{4}}$
Answer Tables How to enter your answer (opens in new window)

Studdy Solution

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 $\frac{1,814,400}{210} = \textbf{8,640}$ .

STEP 12

The final answer is 8,640.

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QuestionEvaluate the following expression. Express your answer as a fraction or a decimal number rounded to four decimal places. $\frac{{ }_{10} P_{8}}{{ }_{10} C_{4}}$
Answer Tables How to enter your answer (opens in new window)

Studdy Solution

STEP 1

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

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

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 $\frac{1,814,400}{210} = \textbf{8,640}$ .

STEP 12

The final answer is 8,640.

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QuestionEvaluate the following expression. Express your answer as a fraction or a decimal number rounded to four decimal places. 10P810C4\frac{{ }_{10} P_{8}}{{ }_{10} C_{4}}10​C4​10​P8​​Answer Tables How to enter your answer (opens in new window)

Studdy Solution

STEP 1

STEP 2

1. Calculate the Permutation2. Calculate the Combination3. Calculate the Fraction

STEP 3

Let's **start** with the **permutation**!We're looking at 10P8_{10}P_810​P8​, 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 nPr=n!(n−r)!_{n}P_r = \frac{n!}{(n-r)!}n​Pr​=(n−r)!n!​.In our case, n=10n = \textbf{10}n=10 and r=8r = \textbf{8}r=8, so we have 10P8=10!(10−8)!_{10}P_8 = \frac{10!}{(10-8)!}10​P8​=(10−8)!10!​.

STEP 5

STEP 6

Now, let's **tackle** the **combination**!We have 10C4_{10}C_410​C4​, 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 nCr=n!r!(n−r)!_{n}C_r = \frac{n!}{r!(n-r)!}n​Cr​=r!(n−r)!n!​.Here, n=10n = \textbf{10}n=10 and r=4r = \textbf{4}r=4, so 10C4=10!4!(10−4)!=10!4!6!_{10}C_4 = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!}10​C4​=4!(10−4)!10!​=4!6!10!​.

STEP 8

STEP 9

We can **simplify** further: 10⋅9⋅8⋅74⋅3⋅2⋅1=504024=210\frac{10 \cdot 9 \cdot 8 \cdot 7}{4 \cdot 3 \cdot 2 \cdot 1} = \frac{5040}{24} = \textbf{210}4⋅3⋅2⋅110⋅9⋅8⋅7​=245040​=210.There are **210** ways to choose 4 items out of 10.

STEP 10

Finally, we need to **calculate** the **fraction**: 10P810C4\frac{_{10}P_8}{_{10}C_4}10​C4​10​P8​​.We already calculated the **permutation** 10P8=1,814,400_{10}P_8 = \textbf{1,814,400}10​P8​=1,814,400 and the **combination** 10C4=210_{10}C_4 = \textbf{210}10​C4​=210.

STEP 11

So, the **fraction** is 1,814,400210=8,640\frac{1,814,400}{210} = \textbf{8,640}2101,814,400​=8,640.

STEP 12

The final answer is **8,640**.

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QuestionEvaluate the following expression. Express your answer as a fraction or a decimal number rounded to four decimal places. $\frac{{ }_{10} P_{8}}{{ }_{10} C_{4}}$
Answer Tables How to enter your answer (opens in new window)

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

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.

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 $_{10}P_8 = \frac{10!}{(10-8)!}$ .

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.

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

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.

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 $_{10}C_4 = \textbf{210}$ .

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

The final answer is 8,640.