Basic Permutations 1

Question 1 of 5

How many arrangements can be made with

$ABC$ ?

(6)

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

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

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

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 letter:

We can choose

$3$ letters:

$A,B$ and

$C$

$=$ $3$

Second letter:

One has already been chosen from

$A,B$ and

$C$ . Hence we are left with

$2$ choices

$=$ $2$

Second spot:

Two have already been chosen from

$A,B$ and

$C$ . Hence we are left with

$1$ choice

$=$ $1$

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

number of ways	$=$	$m$ $times$ $n$	Fundamental Counting Principle
	$=$	$3$ $times$ $2$ $times$ $1$
	$=$	$12$

There are

$6$ ways to arrange

$ABC$

$\color{red}{A}\color{green}{B}\color{blue}{C}$	$\color{red}{A}\color{blue}{C}\color{green}{B}$	$\color{green}{B}\color{red}{A}\color{blue}{C}$
$\color{blue}{C}\color{green}{B}\color{red}{A}$	$\color{green}{B}\color{blue}{C}\color{red}{A}$	$\color{blue}{C}\color{red}{A}\color{green}{B}$

$6$

Method Two

We need to arrange $3$ letters $(r)$ into $3$ positions $(n)$

$r=3$

$n=3$

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

$_\color{purple}{3}P_{\color{green}{3}}$	$=$	$\frac{\color{purple}{3}!}{(\color{purple}{3}-\color{green}{3})!}$	Substitute the values of $n$ and $r$

	$=$	$\frac{3!}{0!}$

	$=$	$3\cdot2\cdot1$	$0! =1$
	$=$	$6$

There are

$6$ ways to arrange

$ABC$

$\color{red}{A}\color{green}{B}\color{blue}{C}$	$\color{red}{A}\color{blue}{C}\color{green}{B}$	$\color{green}{B}\color{red}{A}\color{blue}{C}$
$\color{blue}{C}\color{green}{B}\color{red}{A}$	$\color{green}{B}\color{blue}{C}\color{red}{A}$	$\color{blue}{C}\color{red}{A}\color{green}{B}$

$6$

Question 2 of 5

Evaluate

$_4 P_2$

(12)

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

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.

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

$=$

$_\color{purple}{4}P_{\color{green}{2}}$

$r$	$=$	$2$
$n$	$=$	$4$

Substitute the values into the permutation formula.

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

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

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

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

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

$12$

Question 3 of 5

Evaluate

$_8 P_1$

(8)

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

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.

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

$=$

$_\color{purple}{8}P_{\color{green}{1}}$

$r$	$=$	$1$
$n$	$=$	$8$

Substitute the values into the permutation formula.

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

$_\color{purple}{8}P_{\color{green}{1}}$	$=$	$\frac{\color{purple}{8}!}{(\color{purple}{8}-\color{green}{1})!}$	Substitute the values of $n$ and $r$

	$=$	$\frac{8!}{7!}$

	$=$	$\frac{8\cdot{\color{#CC0000}{7}}\cdot{\color{#CC0000}{6}}\cdot{\color{#CC0000}{5}}\cdot{\color{#CC0000}{4}}\cdot{\color{#CC0000}{3}}\cdot{\color{#CC0000}{2}}\cdot{\color{#CC0000}{1}}}{{\color{#CC0000}{7}}\cdot{\color{#CC0000}{6}}\cdot{\color{#CC0000}{5}}\cdot{\color{#CC0000}{4}}\cdot{\color{#CC0000}{3}}\cdot{\color{#CC0000}{2}}\cdot{\color{#CC0000}{1}}}$

	$=$	$8$	Cancel like terms

$8$

Question 4 of 5

Evaluate

$_7 P_7$

(5040)

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

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.

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

$=$

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

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

Substitute the values into the permutation formula.

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

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

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

	$=$	$\frac{7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{1}$	$0! =1$

	$=$	$5040$

$5040$

Question 5 of 5

How many arrangements can be made with

$\text(MOUSE)$ ?

(120)

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

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

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

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 letter:

We can choose from

$5$ letters:

$M,O,U,S$ and

$E$

$=$ $5$

Second letter:

One has already been chosen from

$M,O,U,S$ and

$E$ . Hence we are left with

$4$ choices

$=$ $4$

Third letter:

Two have already been chosen from

$M,O,U,S$ and

$E$ . Hence we are left with

$3$ choices

$=$ $3$

Fourth letter:

Three have already been chosen from

$M,O,U,S$ and

$E$ . Hence we are left with

$2$ choices

$=$ $2$

Fifth letter:

Four have already been chosen from

$M,O,U,S$ and

$E$ . Hence we are left with

$1$ choice

$=$ $1$

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

$\text(MOUSE)$

$120$

Method Two

We need to arrange $5$ letters $(r)$ 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

$\text(MOUSE)$

$120$

Try VividMath Premium to unlock full access

Quiz summary

Information

1. Question

Hint

Great Work!

Permutation Formula

Fundamental Counting Principle

2. Question

Hint

Correct!

Permutation Formula

3. Question

Hint

Keep Going!

Permutation Formula

4. Question

Hint

Fantastic!

Permutation Formula

5. Question

Hint

Excellent!

Permutation Formula

Fundamental Counting Principle

Quizzes