Basic Permutations 2

Question 1 of 5

Peter and Erica will be checking into a hotel with

$5$ 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 $(r)$ from the total number of items $(n)$ .

Remember that order is important in Permutations.

Permutation Formula

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

Fundamental Counting Principle

number of ways

$=$ $m$

$times$ $n$

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

$5$ vacant rooms

$=$ $5$

Erica’s room:

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

$4$ vacant rooms

$=$ $4$

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

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

There are

$20$ ways to assign

$2$ rooms from

$5$ vacant rooms.

$20$

Method Two

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

$r=2$

$n=5$

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

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

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

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

There are

$20$ ways to assign

$2$ rooms from

$5$ vacant rooms.

$20$

Question 2 of 5

How many ways can we arrange

$5$ tools into a tool shed?

(120)

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

Remember that order is important in Permutations.

Permutation Formula

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

Fundamental Counting Principle

number of ways

$=$ $m$

$times$ $n$

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

$5$ tools

$=$ $5$

Second tool:

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

$4$ tools

$=$ $4$

Third tool:

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

$3$ tools

$=$ $3$

Fourth tool:

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

$2$ tools

$=$ $2$

Fifth tool:

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

$1$ tool

$=$ $1$

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

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

There are

$120$ ways to arrange

$5$ tools.

$120$

Method Two

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

$r=5$

$n=5$

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

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

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

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

There are

$120$ ways to arrange

$5$ tools.

$120$

Question 3 of 5

How many ways can we assign

$7$ nurses as speakers at a conference?

(5040)

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

Remember that order is important in Permutations.

Permutation Formula

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

Fundamental Counting Principle

number of ways

$=$ $m$

$times$ $n$

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

$7$ nurses

$=$ $7$

Second speaker:

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

$6$ nurses

$=$ $6$

Third speaker:

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

$5$ nurses

$=$ $5$

Fourth speaker:

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

$4$ nurses

$=$ $4$

Fifth speaker:

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

$3$ nurses

$=$ $3$

Sixth speaker:

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

$2$ nurses

$=$ $2$

Seventh speaker:

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

$1$ nurse

$=$ $1$

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

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

There are

$5040$ ways to assign

$7$ nurses as speakers at a conference.

$5040$

Method Two

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

$r=7$

$n=7$

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

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

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

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

There are

$5040$ ways to assign

$7$ nurses as speakers at a conference.

$5040$

Question 4 of 5

How many ways can we arrange

$9$ songs on a playlist?

(362880)

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

Remember that order is important in Permutations.

Permutation Formula

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

Fundamental Counting Principle

number of ways

$=$ $m$

$times$ $n$

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

$9$ songs

$=$ $9$

Second song:

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

$8$ songs

$=$ $8$

Third song:

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

$7$ songs

$=$ $7$

Fourth song:

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

$6$ songs

$=$ $6$

Fifth song:

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

$5$ songs

$=$ $5$

Sixth song:

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

$4$ songs

$=$ $4$

Seventh song:

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

$3$ songs

$=$ $3$

Eighth song:

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

$2$ songs

$=$ $2$

Ninth song:

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

$1$ song

$=$ $1$

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

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

There are

$362 880$ ways to arrange

$9$ songs on a playlist.

$362 880$

Method Two

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

$r=9$

$n=9$

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

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

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

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

There are

$362 880$ ways to arrange

$9$ songs on a playlist.

$362 880$

Question 5 of 5

How many ways can we arrange

$8$ books on a shelf that can hold

$4$ books at a time?

(1680)

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

Remember that order is important in Permutations.

Permutation Formula

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

Fundamental Counting Principle

number of ways

$=$ $m$

$times$ $n$

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

$8$ books

$=$ $8$

Second spot:

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

$7$ books

$=$ $7$

Third spot:

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

$6$ books

$=$ $6$

Fourth spot:

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

$5$ books

$=$ $5$

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

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

There are

$1680$ ways to arrange

$8$ books on a shelf that can hold

$4$ books.

$1680$

Method Two

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

$r=4$

$n=8$

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

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

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

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

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

There are

$1680$ ways to arrange

$8$ books on a shelf that can hold

$4$ books.

$1680$

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Quiz summary

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1. Question

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Nice Job!

Permutation Formula

Fundamental Counting Principle

2. Question

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Well Done!

Permutation Formula

Fundamental Counting Principle

3. Question

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Correct!

Permutation Formula

Fundamental Counting Principle

4. Question

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Fantastic!

Permutation Formula

Fundamental Counting Principle

5. Question

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Nice Job!

Permutation Formula

Fundamental Counting Principle

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