Basic Permutations 3

Question 1 of 5

A diner offers

4

$4$ different meals. How many ways can we have

2

$2$ different meals at a time?

(12)

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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

First meal:

We can choose from any of the

4

$4$ meals

=

$=$ $4$ $4$

Second meal:

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

3

$3$ meals

=

$=$ $3$ $3$

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

number of ways	$=$ $=$	$m$ $m$ $\times$ $times$ $n$ $n$	Fundamental Counting Principle
	$=$ $=$	$4$ $4$ $\times$ $times$ $3$ $3$
	$=$ $=$	$12$ $12$

There are

12

$12$ ways to have

2

$2$ different meals.

12

$12$

Method Two

We can only have $2$ $2$ different meals $(r)$ $(r)$ from a menu of $4$ $4$ meals $(n)$ $(n)$

r = 2

$r=2$

n = 4

$n=4$

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

$_\color{purple}{4}P_{\color{green}{2}}$	$=$ $=$	$\frac{\color{purple}{4}!}{(\color{purple}{4}-\color{green}{2})!}$	Substitute the values of $n$ $n$ and $r$ $r$

	$=$ $=$	$\frac{4!}{2!}$

	$=$ $=$	$\frac{4\cdot3\cdot{\color{#CC0000}{2}}\cdot{\color{#CC0000}{1}}}{{\color{#CC0000}{2}}\cdot{\color{#CC0000}{1}}}$

	$=$ $=$	$4\cdot3$	Cancel like terms
	$=$ $=$	$12$ $12$

There are

12

$12$ ways to have

2

$2$ different meals.

12

$12$

Question 2 of 5

How many ways can we select

2

$2$ flags from a set of

7

$7$ different flags?

(42)

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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 flag:

We can choose from any of the

7

$7$ flags

=

$=$ $7$ $7$

Second flag:

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

6

$6$ flags

=

$=$ $6$ $6$

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

number of ways	$=$ $=$	$m$ $m$ $\times$ $times$ $n$ $n$	Fundamental Counting Principle
	$=$ $=$	$7$ $7$ $\times$ $times$ $6$ $6$
	$=$ $=$	$42$ $42$

There are

42

$42$ ways to select

2

$2$ flags from a set of

7

$7$ flags.

42

$42$

Method Two

We need to select $2$ $2$ flags $(r)$ $(r)$ from $7$ $7$ different flags $(n)$ $(n)$

r = 2

$r=2$

n = 7

$n=7$

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

$_\color{purple}{7}P_{\color{green}{2}}$	$=$ $=$	$\frac{\color{purple}{7}!}{(\color{purple}{7}-\color{green}{2})!}$	Substitute the values of $n$ $n$ and $r$ $r$

	$=$ $=$	$\frac{7!}{5!}$

	$=$ $=$	$\frac{7\cdot6\cdot{\color{#CC0000}{5}}\cdot{\color{#CC0000}{4}}\cdot{\color{#CC0000}{3}}\cdot{\color{#CC0000}{2}}\cdot{\color{#CC0000}{1}}}{{\color{#CC0000}{5}}\cdot{\color{#CC0000}{4}}\cdot{\color{#CC0000}{3}}\cdot{\color{#CC0000}{2}}\cdot{\color{#CC0000}{1}}}$

	$=$ $=$	$7\cdot6$	Cancel like terms
	$=$ $=$	$42$ $42$

There are

42

$42$ ways to select

2

$2$ flags from a set of

7

$7$ flags.

42

$42$

Question 3 of 5

How many ways can we select

4

$4$ flags from a set of

7

$7$ different flags?

(840)

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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 flag:

We can choose from any of the

7

$7$ flags

=

$=$ $7$ $7$

Second flag:

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

$6$ flags

$=$ $6$

Third flag:

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

$5$ flags

$=$ $5$

Fourth flag:

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

$4$ flags

$=$ $4$

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

number of ways	$=$	$m$ $times$ $n$	Fundamental Counting Principle
	$=$	$7$ $times$ $6$ $times$ $5$ $times$ $4$
	$=$	$840$

There are

$840$ ways to select

$4$ flags from a set of

$7$ flags.

$840$

We need to select $4$ flags $(r)$ from $7$ different flags $(n)$

$r=4$

$n=7$

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

$_\color{purple}{7}P_{\color{green}{2}}$	$=$	$\frac{\color{purple}{7}!}{(\color{purple}{7}-\color{green}{4})!}$	Substitute the values of $n$ and $r$

	$=$	$\frac{7!}{3!}$

	$=$	$\frac{7\cdot6\cdot5\cdot4\cdot{\color{#CC0000}{3}}\cdot{\color{#CC0000}{2}}\cdot{\color{#CC0000}{1}}}{{\color{#CC0000}{3}}\cdot{\color{#CC0000}{2}}\cdot{\color{#CC0000}{1}}}$

	$=$	$7\cdot6\cdot5\cdot4$	Cancel like terms
	$=$	$840$

There are

$840$ ways to select

$4$ flags from a set of

$7$ flags.

$840$

Question 4 of 5

Find the value of

$r$

$_8 P_r=336$

$r=$ (3)

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Mute

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

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

First, identify the values in the formula.

$_\color{purple}{8}P_{\color{green}{r}}$

$=$

$336$

$r$	$=$	$r$
$n$	$=$	$8$

Substitute the values into the permutation formula.

$_\color{purple}{n}P_{\color{green}{r}}$

$=$

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

$_\color{purple}{8}P_{\color{green}{r}}$	$=$	$336$	Substitute the values of $n$ and $r$

$\frac{\color{purple}{8}!}{(\color{purple}{8}-\color{green}{r})!}$	$=$	$336$	Permutation Formula

$336(8-r)!$	$=$	$8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1$	Cross multiply
$336(8-r)!$ $divide336$	$=$	$40320$ $divide336$	Divide both sides by $336$
$(8-r)!$	$=$	$120$
$(8-r)!$	$=$	$5!$	$120=5xx4xx3xx2xx1$
$8-r$	$=$	$5$	Cancel the factorial from both sides
$8-r$ $-8$	$=$	$5$ $-8$	Subtract $5$ from both sides
$-r$ $times(-1)$	$=$	$-3$ $times(-1)$	Multiply both sides by $-1$
$r$	$=$	$3$

$r=3$

Question 5 of 5

Find the value of

$r$

$_7 P_r=840$

(4)

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Mute

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

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

First, identify the values in the formula.

$_\color{purple}{7}P_{\color{green}{r}}$

$=$

$840$

$r$	$=$	$r$
$n$	$=$	$7$

Substitute the values into the permutation formula.

$_\color{purple}{n}P_{\color{green}{r}}$

$=$

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

$_\color{purple}{7}P_{\color{green}{r}}$	$=$	$840$	Substitute the values of $n$ and $r$

$\frac{\color{purple}{7}!}{(\color{purple}{7}-\color{green}{r})!}$	$=$	$840$	Permutation Formula

$840(7-r)!$	$=$	$7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1$	Cross multiply
$840(7-r)!$ $divide840$	$=$	$5040$ $divide840$	Divide both sides by $840$
$(7-r)!$	$=$	$6$
$(7-r)!$	$=$	$3!$	$6=3xx2xx1$
$7-r$	$=$	$3$	Cancel the factorial from both sides
$7-r$ $-7$	$=$	$3$ $-7$	Subtract $5$ from both sides
$-r$ $times(-1)$	$=$	$-4$ $times(-1)$	Multiply both sides by $-1$
$r$	$=$	$4$

$r=4$

Basic Permutations 3

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Quiz summary

Information

1. Question

Hint

Well Done!

Permutation Formula

Fundamental Counting Principle

2. Question

Hint

Excellent!

Permutation Formula

Fundamental Counting Principle

3. Question

Hint

Keep Going!

Permutation Formula

Fundamental Counting Principle

4. Question

Hint

Great Work!

Permutation Formula

5. Question

Hint

Exceptional!

Permutation Formula

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