Permutation Problems 2

Years

Year 12

Permutations and Combinations

Permutation Problems

Permutation Problems 2

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Question 1 of 5

How many ways can

$8$ paintings be arranged in a row?

(40320)

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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 position in the row

First position:

There are

$8$ paintings to choose from

$=$ $8$

Second position:

From

$8$ paintings, one has already been chosen. Hence, we are left with

$7$ choices

$=$ $7$

Third position:

From

$8$ paintings, two has already been chosen. Hence, we are left with

$6$ choices

$=$ $6$

Fourth position:

From

$8$ paintings, three has already been chosen. Hence, we are left with

$5$ choices

$=$ $5$

Fifth position:

From

$8$ paintings, four has already been chosen. Hence, we are left with

$4$ choices

$=$ $4$

Sixth position:

From

$8$ paintings, five has already been chosen. Hence, we are left with

$3$ choices

$=$ $3$

Seventh position:

From

$8$ paintings, six has already been chosen. Hence, we are left with

$2$ choices

$=$ $2$

Eighth position:

From

$8$ paintings, seven has already been chosen. Hence, we are left with

$1$ choice

$=$ $1$

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

number of ways	$=$	$m$ $times$ $n$	Fundamental Counting Principle
	$=$	$8$ $times$ $7$ $times$ $6$ $times$ $5$ $times$ $4$ $times$ $3$ $times$ $2$ $times$ $1$
	$=$	$40 320$

There are $40 320$ ways to arrange

$8$ paintings

$40 320$

Method Two

We are arranging $8$ paintings $(r)$ for $8$ positions in a row $(n)$

$r=8$

$n=8$

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

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

	$=$	$\frac{8!}{0!}$

	$=$	$\frac{8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{1}$	$0! =1$

	$=$	$40 320$

There are $40 320$ ways to arrange

$8$ paintings

$40 320$

Question 2 of 5

Out of

$8$ paintings, how many ways can

$5$ paintings be chosen to be arranged in a row?

(6720)

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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 position in the row

First position:

There are

$8$ paintings to choose from

$=$ $8$

Second position:

From

$8$ paintings, one has already been chosen. Hence, we are left with

$7$ choices

$=$ $7$

Third position:

From

$8$ paintings, two has already been chosen. Hence, we are left with

$6$ choices

$=$ $6$

Fourth position:

From

$8$ paintings, three has already been chosen. Hence, we are left with

$5$ choices

$=$ $5$

Fifth position:

From

$8$ paintings, four has already been chosen. Hence, we are left with

$4$ choices

$=$ $4$

Use the Fundamental Counting Principle and multiply each draw’s number of options.

number of ways	$=$	$m$ $times$ $n$	Fundamental Counting Principle
	$=$	$8$ $times$ $7$ $times$ $6$ $times$ $5$ $times$ $4$
	$=$	$6720$

There are $6720$ ways to arrange

$5$ paintings out of

$8$

$6720$

Method Two

We are arranging $5$ paintings $(r)$ out of $8$ paintings $(n)$

$r=5$

$n=8$

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

$_\color{purple}{8}P_{\color{green}{8}}$	$=$	$\frac{\color{purple}{8}!}{(\color{purple}{8}-\color{green}{5})!}$	Substitute the value of $n$ and $r$

	$=$	$\frac{8!}{3!}$

	$=$	$\frac{8\cdot7\cdot6\cdot5\cdot4\cdot\color{#CC0000}{3\cdot2\cdot1}}{\color{#CC0000}{3\cdot2\cdot1}}$

	$=$	$6720$	Cancel like terms and evaluate

There are $6720$ ways to arrange

$5$ paintings out of

$8$

$6720$

Question 3 of 5

In a box containing

$8$ cards marked

$1$ -

$8$ , what is the probability of drawing

$345$ in that order?

1. $1/112$
2. $1/56$
3. $1/336$
4. $6/49$

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Probability

$\frac{\color{#e65021}{\mathsf{favourable\:outcome}}}{\color{#007DDC}{\mathsf{total\:outcome}}}$

Permutation Formula

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

First, find the total number of ways we can draw $3$ cards $(r)$ from a box containing $8$ cards $(n)$

$r=3$

$n=8$

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

$_\color{purple}{8}P_{\color{green}{3}}$	$=$	$\frac{\color{purple}{8}!}{(\color{purple}{8}-\color{green}{3})!}$	Substitute the values of $n$ and $r$

	$=$	$\frac{8!}{5!}$

	$=$	$\frac{8\cdot7\cdot6\cdot\color{#CC0000}{5\cdot4\cdot3\cdot2\cdot1}}{\color{#CC0000}{5\cdot4\cdot3\cdot2\cdot1}}$

	$=$	$336$	Cancel like terms and evaluate

There are $336$ ways to draw

$3$ cards from a box containing

$8$ cards. This is the total outcome

Remember that we want to draw the cards

$345$ in that order. This means that the favourable outcome is only $1$ .

Compute for the probability.

Probability	$=$	$\frac{\color{#e65021}{\mathsf{favourable\:outcome}}}{\color{#007DDC}{\mathsf{total\:outcome}}}$

	$=$	$\frac{\color{#e65021}{1}}{\color{#007DDC}{336}}$

The probability of drawing the cards

$345$ in that order is

$1/336$

Question 4 of 5

In a horse race with

$13$ competing horses, what is the probability of winning a Trifecta (successfully selecting the

$1$ st,

$2$ nd and

$3$ rd places)?

1. $2/163$
2. $1/1716$
3. $1/356$
4. $5/1716$

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Probability

$\frac{\color{#e65021}{\mathsf{favourable\:outcome}}}{\color{#007DDC}{\mathsf{total\:outcome}}}$

Permutation Formula

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

First, find the total number of ways we can select $3$ horses $(r)$ out of $13$ $(n)$

$r=3$

$n=13$

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

$_\color{purple}{13}P_{\color{green}{3}}$	$=$	$\frac{\color{purple}{13}!}{(\color{purple}{13}-\color{green}{3})!}$	Substitute the values of $n$ and $r$

	$=$	$\frac{13!}{10!}$

	$=$	$\frac{13\cdot12\cdot11\cdot\color{#CC0000}{10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}}{\color{#CC0000}{10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}}$

	$=$	$1716$	Cancel like terms and evaluate

There are $1716$ ways to select

$3$ horses from

$13$ competing horses in the race. This is the total outcome

Remember that we want to successfully pick the

$1$ st,

$2$ nd and

$3$ rd ranking horse. This means that the favourable outcome is only $1$ .

Compute for the probability.

Probability	$=$	$\frac{\color{#e65021}{\mathsf{favourable\:outcome}}}{\color{#007DDC}{\mathsf{total\:outcome}}}$

	$=$	$\frac{\color{#e65021}{1}}{\color{#007DDC}{1716}}$

The probability of a trifecta is

$1/1716$

Question 5 of 5

5. Question
How many ways can we draw $3$ cards from a box containing $8$ cards marked with $1$ - $8$ , if each card is drawn with replacement?
- (512)
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Fundamental Counting Principle

number of ways $=$ $m$ $times$ $n$

First, count the options for each draw

Remember that the cards are being drawn with replacement

First to Third draw:

There $8$ cards in the box. Since each drawn card is replaced, all three draws have $8$ options

$=$ $8$

Use the Fundamental Counting Principle and multiply each category’s number of options.

number of ways $=$ $m$ $times$ $n$ Fundamental Counting Principle

$=$ $8$ $times$ $8$ $times$ $8$

$=$ $512$

There are $512$ ways of drawing $3$ cards from the box with replacement

$512$

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Quiz summary

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1. Question

Hint

Great Work!

Permutation Formula

Fundamental Counting Principle

2. Question

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Well Done!

Permutation Formula

Fundamental Counting Principle

3. Question

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Excellent!

Probability

Permutation Formula

4. Question

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Keep Going!

Probability

Permutation Formula

5. Question

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Great Work!

Fundamental Counting Principle

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