Permutations with Restrictions 4

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Question 1 of 7

1. Question

Find the number of ways `3` unique cats and `5` unique dogs can be seated in a straight line, given that the `3` cats must always sit together

(4320)

Hint

Help Video

Correct

Correct!

Incorrect

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 if `(n=r)`

$$ _\color{purple}{n}P_{\color{purple}{n}}=\color{purple}{n}! $$

Solve the number of arrangements if the three cats are treated as one entity and the number of arrangements for those `3` cats, then multiply them.

First, treat the `3` cats as one. This leaves us with `6` animals `(r)` to be seated in `6` places in the straight line `(n)`

`n=r=4`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{6}P_{\color{purple}{6}}$$	`=`	$$\color{purple}{6}!$$	Substitute the value of `n`

	`=`	$$6\cdot5\cdot4\cdot3\cdot2\cdot1$$

	`=`	$$720$$

There are `720` ways to arrange `6` animals.

Next, arrange the `3` cats `(r)` in `3` positions `(n)`

`n=r=3`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{3}P_{\color{purple}{3}}$$	`=`	$$\color{purple}{3}!$$	Substitute the value of `n`

	`=`	$$3\cdot2\cdot1$$

	`=`	$$6$$

There are `6` ways to arrange three cats

Finally, multiply the two solved permutations

_`6``P`_`6``=720`

_`3``P`_`3``=6`

$$720\cdot6$$

`=`

$$4320$$

Therefore, there are `4320` ways of arranging `3` cats and `5` dogs if the cats should be seated together

`4320`

Question 2 of 7

2. Question

A bakery has a section for Muffins, Donuts and Cookies. How many ways can `6` muffins, `3` donuts, and `2` cookies be arranged if they must stay in their respective sections?

`51 840`
`4320`
`720`
`8640`

Hint

Help Video

Correct

Keep Going!

Incorrect

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 if `(n=r)`

$$ _\color{purple}{n}P_{\color{purple}{n}}=\color{purple}{n}! $$

Solve and multiply four permutations for: the sections, muffins, donuts and cookies.

First, arrange `3` sections `(r)` in `3` positions `(n)`

`n=r=3`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{3}P_{\color{purple}{3}}$$	`=`	$$\color{purple}{3}!$$	Substitute the value of `n`

	`=`	$$3\cdot2\cdot1$$

	`=`	$$6$$

There are `6` ways to arrange `3` sections.

Next, arrange `6` muffins `(r)` in `6` positions `(n)`

`n=r=6`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{6}P_{\color{purple}{6}}$$	`=`	$$\color{purple}{6}!$$	Substitute the value of `n`

	`=`	$$6\cdot5\cdot4\cdot3\cdot2\cdot1$$

	`=`	$$720$$

There are `720` ways to arrange `6` muffins

Then, arrange `3` donuts `(r)` in `3` positions `(n)`

`n=r=3`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{3}P_{\color{purple}{3}}$$	`=`	$$\color{purple}{3}!$$	Substitute the value of `n`

	`=`	$$3\cdot2\cdot1$$

	`=`	$$6$$

There are `6` ways to arrange `3` donuts

Now, arrange `2` cookies `(r)` in `2` positions `(n)`

`n=r=2`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{2}P_{\color{purple}{2}}$$	`=`	$$\color{purple}{3}!$$	Substitute the value of `n`

	`=`	$$2\cdot1$$

	`=`	$$2$$

There are `2` ways to arrange `2` cookies

Finally, multiply the four solved permutations

sections`=6`

muffins`=720`

donuts`=6`

cookies`=2`

$$6\cdot720\cdot6\cdot2$$

`=`

`51 840`

Therefore, there are `51 840` ways of arranging `6` muffins, `3` donuts and `2` cookies in their respective sections

`51 840`

Question 3 of 7

3. Question

A music playlist has a section for Rock, Jazz and Pop music. How many ways can `4` rock songs, `3` jazz songs, and `2` pop songs be arranged if they must stay in their respective genres?

(1728)

Hint

Help Video

Correct

Fantastic!

Incorrect

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 if `(n=r)`

$$ _\color{purple}{n}P_{\color{purple}{n}}=\color{purple}{n}! $$

Solve and multiply four permutations for: the genres, rock songs, jazz songs and pop songs.

First, arrange `3` genres `(r)` in `3` positions `(n)`

`n=r=3`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{3}P_{\color{purple}{3}}$$	`=`	$$\color{purple}{3}!$$	Substitute the value of `n`

	`=`	$$3\cdot2\cdot1$$

	`=`	$$6$$

There are `6` ways to arrange `3` genres.

Next, arrange `4` rock songs `(r)` in `4` positions `(n)`

`n=r=4`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{4}P_{\color{purple}{4}}$$	`=`	$$\color{purple}{4}!$$	Substitute the value of `n`

	`=`	$$4\cdot3\cdot2\cdot1$$

	`=`	$$24$$

There are `24` ways to arrange `4` rock songs

Then, arrange `3` jazz songs `(r)` in `3` positions `(n)`

`n=r=3`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{3}P_{\color{purple}{3}}$$	`=`	$$\color{purple}{3}!$$	Substitute the value of `n`

	`=`	$$3\cdot2\cdot1$$

	`=`	$$6$$

There are `6` ways to arrange `3` jazz songs

Now, arrange `2` pop songs `(r)` in `2` positions `(n)`

`n=r=2`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{2}P_{\color{purple}{2}}$$	`=`	$$\color{purple}{3}!$$	Substitute the value of `n`

	`=`	$$2\cdot1$$

	`=`	$$2$$

There are `2` ways to arrange `2` pop songs

Finally, multiply the four solved permutations

genres`=6`

rock songs`=24`

jazz songs`=6`

pop songs`=2`

$$6\cdot24\cdot6\cdot2$$

`=`

`1728`

Therefore, there are `1728` ways of arranging `4` rock songs, `3` jazz songs and `2` pop songs in their respective sections

`1728`

Question 4 of 7

4. Question

A music playlist has a section for Rock, Jazz and Pop music. Given that the rock genre must be placed last, how many ways can `4` rock songs, `3` jazz songs, and `2` pop songs be arranged?

(576)

Hint

Help Video

Correct

Excellent!

Incorrect

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 if `(n=r)`

$$ _\color{purple}{n}P_{\color{purple}{n}}=\color{purple}{n}! $$

Solve and multiply four permutations for: the genres, rock songs, jazz songs and pop songs.

First, remember that the rock genre must stay last. This means we are left to arrange `2` genres `(r)` in `2` positions `(n)`

`n=r=2`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{2}P_{\color{purple}{2}}$$	`=`	$$\color{purple}{2}!$$	Substitute the value of `n`

	`=`	$$2\cdot1$$

	`=`	$$2$$

There are `2` ways to arrange `3` genres if rock must stay last.

Next, arrange `4` rock songs `(r)` in `4` positions `(n)`

`n=r=4`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{4}P_{\color{purple}{4}}$$	`=`	$$\color{purple}{4}!$$	Substitute the value of `n`

	`=`	$$4\cdot3\cdot2\cdot1$$

	`=`	$$24$$

There are `24` ways to arrange `4` rock songs

Then, arrange `3` jazz songs `(r)` in `3` positions `(n)`

`n=r=3`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{3}P_{\color{purple}{3}}$$	`=`	$$\color{purple}{3}!$$	Substitute the value of `n`

	`=`	$$3\cdot2\cdot1$$

	`=`	$$6$$

There are `6` ways to arrange `3` jazz songs

Now, arrange `2` pop songs `(r)` in `2` positions `(n)`

`n=r=2`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{2}P_{\color{purple}{2}}$$	`=`	$$\color{purple}{3}!$$	Substitute the value of `n`

	`=`	$$2\cdot1$$

	`=`	$$2$$

There are `2` ways to arrange `2` pop songs

Finally, multiply the four solved permutations

genres`=2`

rock songs`=24`

jazz songs`=6`

pop songs`=2`

$$2\cdot24\cdot6\cdot2$$

`=`

`576`

Therefore, there are `576` ways of arranging `4` rock songs, `3` jazz songs and `2` pop songs if the rock genre must stay last

`576`

Question 5 of 7

5. Question

A pizza booth has `6` seats — `3` on the left side and `3` on the right. How many ways can six people be seated in the booth if `2` girls insist that they sit on the right side?

(144)

Hint

Help Video

Correct

Nice Job!

Incorrect

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

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

Solve the permutation for the two girls who want to sit on the right side and the permutation for the remaining `4` people, then multiply.

First, arrange `2` girls `(r)` in the `3` seats on the right side `(n)`

`n=3`

`r=2`

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

$$_\color{purple}{3}P_{\color{purple}{2}}$$	`=`	$$\frac{\color{purple}{3}!}{(\color{purple}{3}-\color{green}{2})!}$$	Substitute the value of `n` and `r`

	`=`	$$\frac{3!}{1!}$$

	`=`	$$\frac{3\cdot2\cdot\color{#CC0000}{1}}{\color{#CC0000}{1}}$$

	`=`	$$6$$	Cancel like terms and evaluate

There are `6` ways to arrange `2` girls in the three seats on the right.

Next, arrange `4` people `(r)` in `4` seats `(n)`

`n=r=4`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{4}P_{\color{purple}{4}}$$	`=`	$$\color{purple}{4}!$$	Substitute the value of `n`

	`=`	$$4\cdot3\cdot2\cdot1$$

	`=`	$$24$$

There are `24` ways to arrange `4` people

Finally, multiply the two solved permutations

`2` girls`=6`

`4` people`=24`

$$6\cdot24$$

`=`

`144`

Therefore, there are `144` ways of arranging `6` people in the pizza booth if `2` girls want to sit on the right side

`144`

Question 6 of 7

6. Question

A learjet has `8` seats — `4` on the left side and `4` on the right. How many ways can eight people be seated in the learjet if `3` passengers insist to be on the right side and `2` passengers insist to be on the left side?

(1728)

Hint

Help Video

Correct

Well Done!

Incorrect

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

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

Solve the permutation for the `3` passengers who want to sit on the right side, the `2` passengers who want to sit on the left side and the permutation for the remaining `3` people, then multiply.

First, arrange `3` passengers `(r)` in the `4` seats on the right side `(n)`

`n=4`

`r=3`

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

$$_\color{purple}{4}P_{\color{purple}{3}}$$	`=`	$$\frac{\color{purple}{4}!}{(\color{purple}{4}-\color{green}{3})!}$$	Substitute the value of `n` and `r`

	`=`	$$\frac{4!}{1!}$$

	`=`	$$\frac{4\cdot3\cdot2\cdot\color{#CC0000}{1}}{\color{#CC0000}{1}}$$

	`=`	$$24$$	Cancel like terms and evaluate

There are `24` ways to arrange `3` passengers in the four seats on the right.

Next, arrange `2` passengers `(r)` in the `4` seats on the left side `(n)`

`n=4`

`r=2`

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

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

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

	`=`	$$\frac{4\cdot3\cdot\color{#CC0000}{2\cdot1}}{\color{#CC0000}{2\cdot1}}$$

	`=`	$$12$$	Cancel like terms and evaluate

There are `12` ways to arrange `2` passengers in the four seats on the left.

Now, arrange `3` remaining passengers `(r)` in `3` remaining seats `(n)`

`n=r=3`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{3}P_{\color{purple}{3}}$$	`=`	$$\color{purple}{3}!$$	Substitute the value of `n`

	`=`	$$3\cdot2\cdot1$$

	`=`	$$6$$

There are `6` ways to arrange `3` passengers

Finally, multiply the three solved permutations

passengers who want to sit on the right`=24`

passengers who want to sit on the left`=12`

other passengers`=6`

$$24\cdot12\cdot6$$

`=`

`1728`

Therefore, there are `1728` ways of arranging `8` passengers in the learjet if `3` want to sit on the right and `2` want to sit on the left.

`1728`

Question 7 of 7

7. Question

A speedboat has `4` seats — `2` on the port side (left side) and `2` on the right. One girl wishes to sit on the port side. If there are `4` passengers on the boat, what is the probability of that happening?

Write fractions in the format “a/b”

(1/2)

Hint

Help Video

Correct

Exceptional!

Incorrect

Permutation Formula

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

Probability

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

Solve the the number of ways the girl could get a seat on the port side and divide it by the total number of ways all `4` passengers can be arranged.

First, place `1` girl `(r)` in the `2` seats on the port side `(n)`

`n=2`

`r=1`

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

$$_\color{purple}{2}P_{\color{purple}{1}}$$	`=`	$$\frac{\color{purple}{2}!}{(\color{purple}{2}-\color{green}{1})!}$$	Substitute the value of `n` and `r`

	`=`	$$\frac{2!}{1!}$$

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

	`=`	$$2$$	Cancel like terms

There are `2` ways to place `1` girl on the port side.

Now, arrange `3` remaining passengers `(r)` in `3` remaining seats `(n)`

`n=r=3`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{3}P_{\color{purple}{3}}$$	`=`	$$\color{purple}{3}!$$	Substitute the value of `n`

	`=`	$$3\cdot2\cdot1$$

	`=`	$$6$$

There are `6` ways to arrange `3` passengers

Multiply the two solved permutations

girl who wants to sit on the port side`=2`

other passengers`=6`

$$2\cdot6$$

`=`

`12`

Hence, there are `12` ways of arranging `4` passengers on the boat if a girl wants to sit on the port side. This is the favourable outcome

Next, find the total number of ways to arrange all `4` passengers `(r)` in the `4` seats `(n)`

`n=r=4`

$$_\color{purple}{n}P_{\color{purple}{n}}$$	`=`	$$\color{purple}{n}!$$	Permutation Formula (if `n=r`)

$$_\color{purple}{4}P_{\color{purple}{4}}$$	`=`	$$\color{purple}{4}!$$	Substitute the value of `n`

	`=`	$$4\cdot3\cdot2\cdot1$$

	`=`	$$24$$

Hence, there are `24` ways to arrange `4` passengers. This is the total outcome

Finally, solve for the probability

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

	`=`	$$\frac{\color{#e65021}{12}}{\color{#007DDC}{24}}$$

	`=`	$$\frac{1}{2}$$

The probability that the girl gets to sit on the port side is `1/2`

`1/2`

Quizzes

Factorial Notation
Fundamental Counting Principle 1
Fundamental Counting Principle 2
Fundamental Counting Principle 3
Combinations 1
Combinations 2
Combinations with Restrictions 1
Combinations with Restrictions 2
Combinations with Probability
Basic Permutations 1
Basic Permutations 2
Basic Permutations 3
Permutation Problems 1
Permutation Problems 2
Permutations with Repetitions 1
Permutations with Repetitions 2
Permutations with Restrictions 1
Permutations with Restrictions 2
Permutations with Restrictions 3
Permutations with Restrictions 4