Basic Permutations 2

Question 1 of 5

Peter and Erica will be checking into a hotel with

5

$5$ vacant rooms. How many ways can they be assigned to different rooms?

(20)

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Use the permutations formula to find the number of ways an item can be arranged $(r)$ $(r)$ from the total number of items $(n)$ $(n)$ .

Remember that order is important in Permutations.

Permutation Formula

_{n} P_{r} = \frac{n!}{(n - r)!}

$_\color{purple}{n}P_{\color{green}{r}}=\frac{\color{purple}{n}!}{(\color{purple}{n}-\color{green}{r})!}$

Fundamental Counting Principle

number of ways

=

$=$ $m$ $m$

\times

$times$ $n$ $n$

Method One

Solve the problem using the Fundamental Counting Principle

First, count the options for each stage

Peter’s room:

Peter can be assigned to any of the

5

$5$ vacant rooms

=

$=$ $5$ $5$

Erica’s room:

One room has already been assigned to Peter. Hence we are left with

4

$4$ vacant rooms

=

$=$ $4$ $4$

Use the Fundamental Counting Principle and multiply each draws number of options.

number of ways	$=$ $=$	$m$ $m$ $\times$ $times$ $n$ $n$	Fundamental Counting Principle
	$=$ $=$	$5$ $5$ $\times$ $times$ $4$ $4$
	$=$ $=$	$20$ $20$

There are

20

$20$ ways to assign

2

$2$ rooms from

5

$5$ vacant rooms.

20

$20$

Method Two

We need to find how many ways $2$ $2$ rooms $(r)$ $(r)$ can be assigned from $5$ $5$ vacant rooms $(n)$ $(n)$

r = 2

$r=2$

n = 5

$n=5$

$_\color{purple}{n}P_{\color{green}{r}}$	$=$ $=$	$\frac{\color{purple}{n}!}{(\color{purple}{n}-\color{green}{r})!}$	Permutation Formula

$_\color{purple}{5}P_{\color{green}{2}}$	$=$ $=$	$\frac{\color{purple}{5}!}{(\color{purple}{5}-\color{green}{2})!}$	Substitute the values of $n$ $n$ and $r$ $r$

	$=$ $=$	$\frac{5!}{3!}$

	$=$ $=$	$\frac{5\cdot4\cdot{\color{#CC0000}{3}}\cdot{\color{#CC0000}{2}}\cdot{\color{#CC0000}{1}}}{{\color{#CC0000}{3}}\cdot{\color{#CC0000}{2}}\cdot{\color{#CC0000}{1}}}$
	$=$ $=$	$5\cdot4$	Cancel like terms
	$=$ $=$	$20$

There are

20

$20$ ways to assign

2

$2$ rooms from

5

$5$ vacant rooms.

20

$20$

Question 2 of 5

How many ways can we arrange

5

$5$ tools into a tool shed?

(120)

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Mute

Use the permutations formula to find the number of ways an item can be arranged $(r)$ $(r)$ from the total number of items $(n)$ $(n)$ .

Remember that order is important in Permutations.

Permutation Formula

_{n} P_{r} = \frac{n!}{(n - r)!}

$_\color{purple}{n}P_{\color{green}{r}}=\frac{\color{purple}{n}!}{(\color{purple}{n}-\color{green}{r})!}$

Fundamental Counting Principle

number of ways

=

$=$ $m$ $m$

\times

$times$ $n$ $n$

Method One

Solve the problem using the Fundamental Counting Principle

First, count the options for each stage

First tool:

We can choose from any of the

5

$5$ tools

=

$=$ $5$ $5$

Second tool:

One tool has already been chosen. Hence we are left with

4

$4$ tools

=

$=$ $4$ $4$

Third tool:

Two tools have already been chosen. Hence we are left with

3

$3$ tools

=

$=$ $3$ $3$

Fourth tool:

Three tools have already been chosen. Hence we are left with

2

$2$ tools

=

$=$ $2$ $2$

Fifth tool:

Four tools have already been chosen. Hence we are left with

1

$1$ tool

=

$=$ $1$ $1$

Use the Fundamental Counting Principle and multiply each draws number of options.

number of ways	$=$ $=$	$m$ $m$ $\times$ $times$ $n$ $n$	Fundamental Counting Principle
	$=$ $=$	$5$ $5$ $\times$ $times$ $4$ $4$ $\times$ $times$ $3$ $3$ $\times$ $times$ $2$ $2$ $\times$ $times$ $1$ $1$
	$=$ $=$	$120$ $120$

There are

120

$120$ ways to arrange

5

$5$ tools.

120

$120$

Method Two

We need to arrange $5$ $5$ tools $(r)$ $(r)$ into $5$ $5$ positions $(n)$ $(n)$

r = 5

$r=5$

n = 5

$n=5$

$_\color{purple}{n}P_{\color{green}{r}}$	$=$ $=$	$\frac{\color{purple}{n}!}{(\color{purple}{n}-\color{green}{r})!}$	Permutation Formula

$_\color{purple}{5}P_{\color{green}{5}}$	$=$ $=$	$\frac{\color{purple}{5}!}{(\color{purple}{5}-\color{green}{5})!}$	Substitute the values of $n$ $n$ and $r$ $r$

	$=$ $=$	$\frac{5!}{0!}$

	$=$ $=$	$5\cdot4\cdot3\cdot2\cdot1$	$0! = 1$ $0! =1$
	$=$ $=$	$120$

There are

120

$120$ ways to arrange

5

$5$ tools.

120

$120$

Question 3 of 5

How many ways can we assign

7

$7$ nurses as speakers at a conference?

(5040)

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Mute

Use the permutations formula to find the number of ways an item can be arranged $(r)$ $(r)$ from the total number of items $(n)$ $(n)$ .

Remember that order is important in Permutations.

Permutation Formula

_{n} P_{r} = \frac{n!}{(n - r)!}

$_\color{purple}{n}P_{\color{green}{r}}=\frac{\color{purple}{n}!}{(\color{purple}{n}-\color{green}{r})!}$

Fundamental Counting Principle

number of ways

=

$=$ $m$ $m$

\times

$times$ $n$ $n$

Method One

Solve the problem using the Fundamental Counting Principle

First, count the options for each stage

First speaker:

We can choose from any of the

7

$7$ nurses

=

$=$ $7$ $7$

Second speaker:

One nurse has already been chosen. Hence we are left with

6

$6$ nurses

=

$=$ $6$ $6$

Third speaker:

Two nurses have already been chosen. Hence we are left with

5

$5$ nurses

=

$=$ $5$ $5$

Fourth speaker:

Three nurses have already been chosen. Hence we are left with

4

$4$ nurses

=

$=$ $4$ $4$

Fifth speaker:

Four nurses have already been chosen. Hence we are left with

3

$3$ nurses

=

$=$ $3$ $3$

Sixth speaker:

Five nurses have already been chosen. Hence we are left with

2

$2$ nurses

=

$=$ $2$ $2$

Seventh speaker:

Six nurses have already been chosen. Hence we are left with

1

$1$ nurse

=

$=$ $1$ $1$

Use the Fundamental Counting Principle and multiply each draws number of options.

number of ways	$=$ $=$	$m$ $m$ $\times$ $times$ $n$ $n$	Fundamental Counting Principle
	$=$ $=$	$7$ $7$ $\times$ $times$ $6$ $6$ $\times$ $times$ $5$ $5$ $\times$ $times$ $4$ $4$ $\times$ $times$ $3$ $3$ $\times$ $times$ $2$ $2$ $\times$ $times$ $1$ $1$
	$=$ $=$	$5040$ $5040$

There are

5040

$5040$ ways to assign

7

$7$ nurses as speakers at a conference.

5040

$5040$

Method Two

We need to assign $7$ $7$ nurses $(r)$ $(r)$ into $7$ $7$ positions $(n)$ $(n)$

r = 7

$r=7$

n = 7

$n=7$

$_\color{purple}{n}P_{\color{green}{r}}$	$=$ $=$	$\frac{\color{purple}{n}!}{(\color{purple}{n}-\color{green}{r})!}$	Permutation Formula

$_\color{purple}{7}P_{\color{green}{7}}$	$=$ $=$	$\frac{\color{purple}{7}!}{(\color{purple}{7}-\color{green}{7})!}$	Substitute the values of $n$ $n$ and $r$ $r$

	$=$ $=$	$\frac{7!}{0!}$

	$=$ $=$	$7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1$	$0! = 1$ $0! =1$
	$=$ $=$	$5040$

There are

5040

$5040$ ways to assign

7

$7$ nurses as speakers at a conference.

5040

$5040$

Question 4 of 5

How many ways can we arrange

9

$9$ songs on a playlist?

(362880)

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Mute

Use the permutations formula to find the number of ways an item can be arranged $(r)$ $(r)$ from the total number of items $(n)$ $(n)$ .

Remember that order is important in Permutations.

Permutation Formula

_{n} P_{r} = \frac{n!}{(n - r)!}

$_\color{purple}{n}P_{\color{green}{r}}=\frac{\color{purple}{n}!}{(\color{purple}{n}-\color{green}{r})!}$

Fundamental Counting Principle

number of ways

=

$=$ $m$ $m$

\times

$times$ $n$ $n$

Method One

Solve the problem using the Fundamental Counting Principle

First, count the options for each stage

First song:

We can choose from any of the

9

$9$ songs

=

$=$ $9$ $9$

Second song:

One song has already been chosen. Hence we are left with

8

$8$ songs

=

$=$ $8$ $8$

Third song:

Two songs have already been chosen. Hence we are left with

7

$7$ songs

=

$=$ $7$ $7$

Fourth song:

Three songs have already been chosen. Hence we are left with

6

$6$ songs

=

$=$ $6$ $6$

Fifth song:

Four songs have already been chosen. Hence we are left with

5

$5$ songs

=

$=$ $5$ $5$

Sixth song:

Five songs have already been chosen. Hence we are left with

4

$4$ songs

=

$=$ $4$ $4$

Seventh song:

Six songs have already been chosen. Hence we are left with

3

$3$ songs

=

$=$ $3$ $3$

Eighth song:

Seven songs have already been chosen. Hence we are left with

2

$2$ songs

=

$=$ $2$ $2$

Ninth song:

Eight songs have already been chosen. Hence we are left with

1

$1$ song

=

$=$ $1$ $1$

Use the Fundamental Counting Principle and multiply each draws number of options.

number of ways	$=$ $=$	$m$ $m$ $\times$ $times$ $n$ $n$	Fundamental Counting Principle
	$=$ $=$	$9$ $9$ $\times$ $times$ $8$ $8$ $\times$ $times$ $7$ $7$ $\times$ $times$ $6$ $6$ $\times$ $times$ $5$ $5$ $\times$ $times$ $4$ $4$ $\times$ $times$ $3$ $3$ $\times$ $times$ $2$ $2$ $\times$ $times$ $1$ $1$
	$=$ $=$	$362880$ $362 880$

There are

362880

$362 880$ ways to arrange

9

$9$ songs on a playlist.

362880

$362 880$

Method Two

We need to arrange $9$ $9$ songs $(r)$ $(r)$ into $9$ $9$ positions $(n)$ $(n)$

r = 9

$r=9$

n = 9

$n=9$

$_\color{purple}{n}P_{\color{green}{r}}$	$=$ $=$	$\frac{\color{purple}{n}!}{(\color{purple}{n}-\color{green}{r})!}$	Permutation Formula

$_\color{purple}{9}P_{\color{green}{9}}$	$=$ $=$	$\frac{\color{purple}{9}!}{(\color{purple}{9}-\color{green}{9})!}$	Substitute the values of $n$ $n$ and $r$ $r$

	$=$ $=$	$\frac{9!}{0!}$

	$=$ $=$	$9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1$	$0! = 1$ $0! =1$
	$=$ $=$	$362880$ $362 880$

There are

362880

$362 880$ ways to arrange

9

$9$ songs on a playlist.

362880

$362 880$

Question 5 of 5

How many ways can we arrange

8

$8$ books on a shelf that can hold

4

$4$ books at a time?

(1680)

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Mute

Use the permutations formula to find the number of ways an item can be arranged $(r)$ $(r)$ from the total number of items $(n)$ $(n)$ .

Remember that order is important in Permutations.

Permutation Formula

_{n} P_{r} = \frac{n!}{(n - r)!}

$_\color{purple}{n}P_{\color{green}{r}}=\frac{\color{purple}{n}!}{(\color{purple}{n}-\color{green}{r})!}$

Fundamental Counting Principle

number of ways

=

$=$ $m$ $m$

\times

$times$ $n$ $n$

Method One

Solve the problem using the Fundamental Counting Principle

First, count the options for each stage

First spot:

We can choose from any of the

8

$8$ books

=

$=$ $8$ $8$

Second spot:

One book has already been chosen. Hence we are left with

7

$7$ books

=

$=$ $7$ $7$

Third spot:

Two books have already been chosen. Hence we are left with

6

$6$ books

=

$=$ $6$ $6$

Fourth spot:

Three books have already been chosen. Hence we are left with

5

$5$ books

=

$=$ $5$ $5$

Use the Fundamental Counting Principle and multiply each draws number of options.

number of ways	$=$ $=$	$m$ $m$ $\times$ $times$ $n$ $n$	Fundamental Counting Principle
	$=$ $=$	$8$ $8$ $\times$ $times$ $7$ $7$ $\times$ $times$ $6$ $6$ $\times$ $times$ $5$ $5$
	$=$ $=$	$1680$ $1680$

There are

1680

$1680$ ways to arrange

8

$8$ books on a shelf that can hold

4

$4$ books.

1680

$1680$

Method Two

We need to arrange $8$ $8$ books $(r)$ $(r)$ into $4$ $4$ positions $(n)$ $(n)$

r = 4

$r=4$

n = 8

$n=8$

$_\color{purple}{n}P_{\color{green}{r}}$	$=$ $=$	$\frac{\color{purple}{n}!}{(\color{purple}{n}-\color{green}{r})!}$	Permutation Formula

$_\color{purple}{8}P_{\color{green}{4}}$	$=$ $=$	$\frac{\color{purple}{8}!}{(\color{purple}{8}-\color{green}{4})!}$	Substitute the values of $n$ $n$ and $r$ $r$

	$=$ $=$	$\frac{8!}{4!}$

	$=$ $=$	$\frac{8\cdot7\cdot6\cdot5\cdot{\color{#CC0000}{4}}\cdot{\color{#CC0000}{3}}\cdot{\color{#CC0000}{2}}\cdot{\color{#CC0000}{1}}}{{\color{#CC0000}{4}}\cdot{\color{#CC0000}{3}}\cdot{\color{#CC0000}{2}}\cdot{\color{#CC0000}{1}}}$

	$=$ $=$	$8\cdot7\cdot6\cdot5$	Cancel like terms
	$=$ $=$	$1680$ $1680$

There are

1680

$1680$ ways to arrange

8

$8$ books on a shelf that can hold

4

$4$ books.

1680

$1680$

Basic Permutations 2

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Quiz summary

Information

1. Question

Hint

Nice Job!

Permutation Formula

Fundamental Counting Principle

2. Question

Hint

Well Done!

Permutation Formula

Fundamental Counting Principle

3. Question

Hint

Correct!

Permutation Formula

Fundamental Counting Principle

4. Question

Hint

Fantastic!

Permutation Formula

Fundamental Counting Principle

5. Question

Hint

Nice Job!

Permutation Formula

Fundamental Counting Principle

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