Permutations with Restrictions 4

Question 1 of 7

Find the number of ways

3

$3$ unique cats and

5

$5$ unique dogs can be seated in a straight line, given that the

3

$3$ cats must always sit together

(4320)

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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 if $(n = r)$ $(n=r)$

_{n} P_{n} = n!

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

Solve the number of arrangements if the three cats are treated as one entity and the number of arrangements for those

3

$3$ cats, then multiply them.

First, treat the

3

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

n = r = 4

$n=r=4$

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

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

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

	$=$ $=$	$720$

There are

720

$720$ ways to arrange

6

$6$ animals.

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

n = r = 3

$n=r=3$

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

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

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

	$=$ $=$	$6$

There are

6

$6$ ways to arrange three cats

Finally, multiply the two solved permutations

_{$6$ $6$}

P

$P$ _{$6$ $6$}

= 720

$=720$

_{$3$ $3$}

P

$P$ _{$3$ $3$}

= 6

$=6$

720 \cdot 6

$720\cdot6$

=

$=$

4320

$4320$

Therefore, there are $4320$ $4320$ ways of arranging

3

$3$ cats and

5

$5$ dogs if the cats should be seated together

4320

$4320$

Question 2 of 7

A bakery has a section for Muffins, Donuts and Cookies. How many ways can

6

$6$ muffins,

3

$3$ donuts, and

2

$2$ cookies be arranged if they must stay in their respective sections?

1. $51840$ $51 840$
2. $8640$ $8640$
3. $720$ $720$
4. $4320$ $4320$

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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 if $(n = r)$ $(n=r)$

_{n} P_{n} = n!

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

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

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

n = r = 3

$n=r=3$

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

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

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

	$=$ $=$	$6$

There are

6

$6$ ways to arrange

3

$3$ sections.

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

n = r = 6

$n=r=6$

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

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

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

	$=$ $=$	$720$

There are

720

$720$ ways to arrange

6

$6$ muffins

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

n = r = 3

$n=r=3$

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

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

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

	$=$ $=$	$6$

There are

6

$6$ ways to arrange

3

$3$ donuts

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

n = r = 2

$n=r=2$

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

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

	$=$ $=$	$2\cdot1$

	$=$ $=$	$2$

There are

2

$2$ ways to arrange

2

$2$ cookies

Finally, multiply the four solved permutations

sections

= 6

$=6$

muffins

= 720

$=720$

donuts

= 6

$=6$

cookies

= 2

$=2$

6 \cdot 720 \cdot 6 \cdot 2

$6\cdot720\cdot6\cdot2$

=

$=$

51840

$51 840$

Therefore, there are $51840$ $51 840$ ways of arranging

6

$6$ muffins,

3

$3$ donuts and

2

$2$ cookies in their respective sections

51840

$51 840$

Question 3 of 7

A music playlist has a section for Rock, Jazz and Pop music. How many ways can

4

$4$ rock songs,

3

$3$ jazz songs, and

2

$2$ pop songs be arranged if they must stay in their respective genres?

(1728)

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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 if $(n = r)$ $(n=r)$

_{n} P_{n} = n!

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

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

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

n = r = 3

$n=r=3$

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

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

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

	$=$ $=$	$6$

There are

6

$6$ ways to arrange

3

$3$ genres.

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

n = r = 4

$n=r=4$

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

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

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

	$=$ $=$	$24$

There are

24

$24$ ways to arrange

4

$4$ rock songs

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

n = r = 3

$n=r=3$

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

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

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

	$=$ $=$	$6$

There are

6

$6$ ways to arrange

3

$3$ jazz songs

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

n = r = 2

$n=r=2$

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

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

	$=$ $=$	$2\cdot1$

	$=$ $=$	$2$

There are

2

$2$ ways to arrange

2

$2$ pop songs

Finally, multiply the four solved permutations

genres

= 6

$=6$

rock songs

= 24

$=24$

jazz songs

= 6

$=6$

pop songs

= 2

$=2$

6 \cdot 24 \cdot 6 \cdot 2

$6\cdot24\cdot6\cdot2$

=

$=$

1728

$1728$

Therefore, there are $1728$ $1728$ ways of arranging

4

$4$ rock songs,

3

$3$ jazz songs and

2

$2$ pop songs in their respective sections

1728

$1728$

Question 4 of 7

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

$4$ rock songs,

3

$3$ jazz songs, and

2

$2$ pop songs be arranged?

(576)

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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 if $(n = r)$ $(n=r)$

_{n} P_{n} = n!

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

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

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)

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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

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

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

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)

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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

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

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

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)

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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

$1/2$

Permutations with Restrictions 4

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Quiz summary

Information

1. Question

Hint

Correct!

Permutation Formula if $(n = r)$ $(n=r)$

2. Question

Hint

Keep Going!

Permutation Formula if $(n = r)$ $(n=r)$

3. Question

Hint

Fantastic!

Permutation Formula if $(n = r)$ $(n=r)$

4. Question

Hint

Excellent!

Permutation Formula if $(n = r)$ $(n=r)$

5. Question

Hint

Nice Job!

Permutation Formula

6. Question

Hint

Well Done!

Permutation Formula

7. Question

Hint

Exceptional!

Permutation Formula

Probability

Quizzes

Permutations with Restrictions 4

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Quiz summary

Information

1. Question

Hint

Correct!

Permutation Formula if (n=r)(n=r)(n=r)

2. Question

Hint

Keep Going!

Permutation Formula if (n=r)(n=r)(n=r)

3. Question

Hint

Fantastic!

Permutation Formula if (n=r)(n=r)(n=r)

4. Question

Hint

Excellent!

Permutation Formula if (n=r)(n=r)(n=r)

5. Question

Hint

Nice Job!

Permutation Formula

6. Question

Hint

Well Done!

Permutation Formula

7. Question

Hint

Exceptional!

Permutation Formula

Probability

Quizzes

Permutation Formula if $(n = r)$ $(n=r)$

Permutation Formula if $(n = r)$ $(n=r)$

Permutation Formula if $(n = r)$ $(n=r)$

Permutation Formula if $(n = r)$ $(n=r)$