Math / Discrete

QuestionA coordinator will select 8 songs from a list of 9 songs to compose an event's musical entertainment lineup. How many different lineups are possible?
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Studdy Solution

STEP 1

What is this asking? How many different ways can we pick 8 songs out of 9 to make a playlist, where the order of the songs matters? Watch out! The order of songs does matter here, so it's not a simple combination!
We're talking about a lineup, which means the order makes a difference.

STEP 2

1. Permutations

STEP 3

We're looking for the number of permutations, which means how many ways we can arrange a smaller group from a larger group, where order matters.
Think of it like picking the first song, then the second, and so on.

STEP 4

We have 9 choices for the first song.
After picking that one, we have 8 choices left for the second song.
Then 7 choices for the third, and so on, until we've picked 8 songs.

STEP 5

To get the total number of lineups, we multiply these numbers together.
This is because each choice of the first song can be paired with any choice of the second song, and so on.
This gives us $9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2$ .
This is called a permutation.

STEP 6

We can write this using the permutation formula: $P(n, k) = \frac{n!}{(n-k)!}$ , where $n$ is the total number of songs and $k$ is the number of songs we're choosing.
In our case, $n = 9$ and $k = 8$ .

STEP 7

Let's plug in the numbers: $P(9, 8) = \frac{9!}{(9-8)!} = \frac{9!}{1!} = 9!$ .

STEP 8

$9! = 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = \textbf{362,880}$ .

STEP 9

There are a whopping 362,880 different possible lineups!

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QuestionA coordinator will select 8 songs from a list of 9 songs to compose an event's musical entertainment lineup. How many different lineups are possible?
Answer Tables Keypad How to enter your answer (opens in new window) Keyboard Shortcuts

Studdy Solution

STEP 1

What is this asking? How many different ways can we pick 8 songs out of 9 to make a playlist, where the order of the songs matters? Watch out! The order of songs does matter here, so it's not a simple combination!
We're talking about a lineup, which means the order makes a difference.

STEP 2

1. Permutations

STEP 3

We're looking for the number of permutations, which means how many ways we can arrange a smaller group from a larger group, where order matters.
Think of it like picking the first song, then the second, and so on.

STEP 4

We have 9 choices for the first song.
After picking that one, we have 8 choices left for the second song.
Then 7 choices for the third, and so on, until we've picked 8 songs.

STEP 5

To get the total number of lineups, we multiply these numbers together.
This is because each choice of the first song can be paired with any choice of the second song, and so on.
This gives us $9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2$ .
This is called a permutation.

STEP 6

We can write this using the permutation formula: $P(n, k) = \frac{n!}{(n-k)!}$ , where $n$ is the total number of songs and $k$ is the number of songs we're choosing.
In our case, $n = 9$ and $k = 8$ .

STEP 7

Let's plug in the numbers: $P(9, 8) = \frac{9!}{(9-8)!} = \frac{9!}{1!} = 9!$ .

STEP 8

$9! = 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = \textbf{362,880}$ .

STEP 9

There are a whopping 362,880 different possible lineups!

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QuestionA coordinator will select 8 songs from a list of 9 songs to compose an event's musical entertainment lineup. How many different lineups are possible?Answer Tables Keypad How to enter your answer (opens in new window) Keyboard Shortcuts

Studdy Solution

STEP 1

What is this asking? How many different ways can we pick 8 songs out of 9 to make a playlist, where the order of the songs matters? Watch out! The order of songs *does* matter here, so it's not a simple combination!We're talking about a *lineup*, which means the order makes a difference.

STEP 2

1. Permutations

STEP 3

We're looking for the number of *permutations*, which means how many ways we can arrange a smaller group from a larger group, where order matters.Think of it like picking the first song, then the second, and so on.

STEP 4

We have 9 choices for the first song.After picking that one, we have 8 choices left for the second song.Then 7 choices for the third, and so on, until we've picked 8 songs.

STEP 5

STEP 6

We can write this using the **permutation formula**: P(n,k)=n!(n−k)!P(n, k) = \frac{n!}{(n-k)!}P(n,k)=(n−k)!n!​, where nnn is the **total number of songs** and kkk is the **number of songs we're choosing**.In our case, n=9n = 9n=9 and k=8k = 8k=8.

STEP 7

Let's **plug in the numbers**: P(9,8)=9!(9−8)!=9!1!=9!P(9, 8) = \frac{9!}{(9-8)!} = \frac{9!}{1!} = 9!P(9,8)=(9−8)!9!​=1!9!​=9!.

STEP 8

9!=9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1=362,8809! = 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = \textbf{362,880}9!=9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1=362,880.

STEP 9

There are a whopping **362,880** different possible lineups!

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionA coordinator will select 8 songs from a list of 9 songs to compose an event's musical entertainment lineup. How many different lineups are possible?
Answer Tables Keypad How to enter your answer (opens in new window) Keyboard Shortcuts

What is this asking? How many different ways can we pick 8 songs out of 9 to make a playlist, where the order of the songs matters? Watch out! The order of songs does matter here, so it's not a simple combination!
We're talking about a lineup, which means the order makes a difference.

We're looking for the number of permutations, which means how many ways we can arrange a smaller group from a larger group, where order matters.
Think of it like picking the first song, then the second, and so on.

We have 9 choices for the first song.
After picking that one, we have 8 choices left for the second song.
Then 7 choices for the third, and so on, until we've picked 8 songs.

We can write this using the permutation formula: $P(n, k) = \frac{n!}{(n-k)!}$ , where $n$ is the total number of songs and $k$ is the number of songs we're choosing.
In our case, $n = 9$ and $k = 8$ .

Let's plug in the numbers: $P(9, 8) = \frac{9!}{(9-8)!} = \frac{9!}{1!} = 9!$ .

$9! = 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = \textbf{362,880}$ .

There are a whopping 362,880 different possible lineups!