Section 3.1.5: Permutations and Combinations · Combination. 4 Example (1) ... This is a...

Section 3.1.5: Permutationsand Combinations

When picking items from a set sothat the order of the selectionmatters, we call this a Permutation.

When picking items from a set sothat the order of the selection doesnot matter, we call this aCombination.

Example (1)

A club has an election for President,Vice-president and Secretary.

I There are 5 candidates, A, B, C, Dand E

I After ranking, the top candidatewill be President, the secondVice-president and the thirdSecretary.

How many different outcomes arethere?

Example (1)

Example (2)

A club has an election for itsexecutive board, which consists of 3members.

I The top three candidates will beon the board

How many different possibilities arethere for the board?

Example (2)

In both cases, 3 out of the 5candidates will be picked.

In Example (1) the order matterssince it determines who gets whichposition. This is a permutationproblem.

In Example (1) the order matterssince it determines who gets whichposition.

This is a permutationproblem.

In Example (1) the order matterssince it determines who gets whichposition. This is a permutationproblem.

The number of different outcomes isdenoted

In general, if we want to select ritems out of n, and the ordermatters, then the number of waysof doing that is denoted by

The number of different outcomes isdenoted

In general, if we want to select ritems out of n, and the ordermatters, then the number of waysof doing that is denoted by

In Example (2), the order does notmatter since the board members areequal. This is a combinationproblem.

The number of different boards isdenoted by

In general, if we want to select ritems out of n, and the order doesnot matter, then the number ofways of doing that is denoted by

The number of different boards isdenoted by

In general, if we want to select ritems out of n, and the order doesnot matter, then the number ofways of doing that is denoted by

To write a permutation we usesquare brackets like [A,B,C].

This isdifferent from [A,C,B] since the ordermatters.

[A,B,C ] 6= [A,C ,B]

To write a permutation we usesquare brackets like [A,B,C]. This isdifferent from [A,C,B] since the ordermatters.

[A,B,C ] 6= [A,C ,B]

To write a combination, we usebraces like {A,B,C}.

Now {A,B,C}and {A,C,B} are equal since theorder doesn’t matter.

{A,B,C} = {A,C ,B}

To write a combination, we usebraces like {A,B,C}.Now {A,B,C}and {A,C,B} are equal since theorder doesn’t matter.

{A,B,C} = {A,C ,B}

Solution (of Example (1))

Picking the officers can be brokeninto three steps:

1. Pick a President2. Pick a Vice-president3. Pick a Secretary

Picking the officers can be brokeninto three steps:1. Pick a President

2. Pick a Vice-president3. Pick a Secretary

Picking the officers can be brokeninto three steps:1. Pick a President2. Pick a Vice-president

3. Pick a Secretary

Picking the officers can be brokeninto three steps:1. Pick a President2. Pick a Vice-president3. Pick a Secretary

Solution

Remarks:

I The selection of president willaffect the choices forvice-president, since you can’t beboth.

I However, the number of choicesfor vice-president is independentof the selection of president.

Solution

Remarks:

I The selection of president willaffect the choices forvice-president, since you can’t beboth.

I However, the number of choicesfor vice-president is independentof the selection of president.

Solution

I Number of choices for secretary isindependent of the two previoussteps.

I So we can apply the productprinciple to get the total numberof outcomes.

Solution

I Number of choices for secretary isindependent of the two previoussteps.

I So we can apply the productprinciple to get the total numberof outcomes.

President5 ×

Vice-pres

4 ×Secretary

3 =Slates

We see that

5P3 = 5× 4× 3

We go down by 1 since at each stepwe have 1 fewer item to pick from.

In general, if we want to pick r itemsout of n, and the order matters,the product principle tells us that

nPr = n × (n − 1) · · · × (n − r + 1)︸︷︷︸r factors

Special case: When we want to pick nitems out of n and the order matters.

This is the number of ways n itemscan be ordered.

nPn = n×(n−1)×(n−2) · · ·×2×1

Special case: When we want to pick nitems out of n and the order matters.This is the number of ways n itemscan be ordered.

nPn = n×(n−1)×(n−2) · · ·×2×1

We denote this number by n!, andcall it n factorial.

This is the number of permutationsof n objects.

We denote this number by n!, andcall it n factorial.This is the number of permutationsof n objects.

Example

How many ways can we order thethree letters A, B, and C?

Solution

In terms of factorials, it’s

3! = 3× 2× 1 = 6

We can list all six:[A,B,C], [A,C,B], [B,A,C], [B,C,A],[C,A,B], [C,B,A].

Example

Solution

3! = 3× 2× 1 = 6

Example

Solution

3! = 3× 2× 1 = 6

Using factorials we rewrite theformula for nPr as:

nPr =n!

(n − r)!

Now consider combinations.

Example

A club has an election for itsexecutive board, consisting of 2members (different than earlierexample).There are 5 candidates, A,B, C, D and E. The top two will beon the board.How many outcomesare there?

Now consider combinations.Example

A club has an election for itsexecutive board, consisting of 2members (different than earlierexample).

There are 5 candidates, A,B, C, D and E. The top two will beon the board.How many outcomesare there?

A club has an election for itsexecutive board, consisting of 2members (different than earlierexample).There are 5 candidates, A,B, C, D and E.

The top two will beon the board.How many outcomesare there?

A club has an election for itsexecutive board, consisting of 2members (different than earlierexample).There are 5 candidates, A,B, C, D and E. The top two will beon the board.

How many outcomesare there?

A club has an election for itsexecutive board, consisting of 2members (different than earlierexample).There are 5 candidates, A,B, C, D and E. The top two will beon the board.How many outcomesare there?

Solution

We need to choose 2 board membersout of 5 candidates. So the numberof outcomes is

But how do we compute thatnumber?

Solution

We need to choose 2 board membersout of 5 candidates. So the numberof outcomes is

But how do we compute thatnumber?

Solution

If we pretend for now that ordermatters we would get

5P2 = 5× 4 = 20

But outcomes like [A,B] and [B,A](which are different permutations)represent the same combination,namely {A,B}.

Solution

If we pretend for now that ordermatters we would get

5P2 = 5× 4 = 20

But outcomes like [A,B] and [B,A](which are different permutations)represent the same combination,namely {A,B}.

Solution

So, in the above count of 20, eachcombination gets double counted.So, divide by 2, to get

5C2 =5× 4

In general, when r items are pickedthey can be arranged in

rPr = r !

different orders.

So, when we ignore the order, eachcombination gets counted r ! times.

In general, when r items are pickedthey can be arranged in

rPr = r !

different orders.So, when we ignore the order, eachcombination gets counted r ! times.

Therefore the number ofcombinations of r items out of n isobtained by dividing the number ofpermutations by r !:

nCr =nPr

(n − r)!× r !or

nCr =n · (n − 1) · · · (n − r + 1)

1 · 2 · 3 · · · r

nCr =nPr

(n − r)!× r !

nCr =n · (n − 1) · · · (n − r + 1)

1 · 2 · 3 · · · r

nCr =nPr

(n − r)!× r !or

nCr =n · (n − 1) · · · (n − r + 1)

1 · 2 · 3 · · · r

Example

= 6 · 5 · 4 = 120

6C3 =6P3

6 · 5 · 41 · 2 · 3

Example

6P3 = 6 · 5 · 4

6C3 =6P3

6 · 5 · 41 · 2 · 3

Example

6P3 = 6 · 5 · 4 = 120

6C3 =6P3

6 · 5 · 41 · 2 · 3

Example

6P3 = 6 · 5 · 4 = 120

6 · 5 · 41 · 2 · 3

Example

6P3 = 6 · 5 · 4 = 120

6C3 =6P3

=6 · 5 · 41 · 2 · 3

Example

6P3 = 6 · 5 · 4 = 120

6C3 =6P3

6 · 5 · 41 · 2 · 3

Example

6P3 = 6 · 5 · 4 = 120

6C3 =6P3

6 · 5 · 41 · 2 · 3

What if r = 0 or r = n? We need toknow what to plug in for 0!.

Turns out 0! = 1.

n! = n × (n − 1) · · · × 2× 10! = 1

What if r = 0 or r = n? We need toknow what to plug in for 0!.Turns out 0! = 1.

n! = n × (n − 1) · · · × 2× 1

0! = 1

What if r = 0 or r = n? We need toknow what to plug in for 0!.Turns out 0! = 1.

n! = n × (n − 1) · · · × 2× 10! = 1

Example

(6− 0)!= 1

6C6 =6!

6! · 0!= 1

Example

6P0 =6!

(6− 0)!= 1

6C6 =6!

6! · 0!= 1

Example

6P0 =6!

(6− 0)!= 1

6! · 0!= 1

Example

6P0 =6!

(6− 0)!= 1

6C6 =6!

6! · 0!

Example

6P0 =6!

(6− 0)!= 1

6C6 =6!

6! · 0!= 1

Now we have the tools to tackle the“3” version:

Example (2)

A club has an election for itsexecutive board, consisting of 3members.

I Top three candidates will be onthe board

How many outcomes are there?

Now we have the tools to tackle the“3” version:Example (2)

Solution

If 3 board members are chosen froma pool of 5 candidates, this can bedone in

5C3 =5!

2! 3!=

2 · 6= 10

different ways.

Example

An Ice Cream Parlor has 7 flavors ofice cream, in three differentcontainers.

a) How many different two-scooporders are possible?

b) How many true double scoops arepossible?

A true double is a two-scoop with 2different flavors.

Example

An Ice Cream Parlor has 7 flavors ofice cream, in three differentcontainers.a) How many different two-scoop

orders are possible?

b) How many true double scoops arepossible?

Example

orders are possible?b) How many true double scoops are

possible?

Example

orders are possible?b) How many true double scoops are

possible?A true double is a two-scoop with 2different flavors.

First consider the true doubles.

We assume that the order doesn’tmatter. There are

7C2 =7!

2 · 120= 21

ways to choose the two flavors.

First consider the true doubles.We assume that the order doesn’tmatter. There are

7C2 =7!

2 · 120= 21

ways to choose the two flavors.

Now we break down the wholeprocess in two independent steps:

I pick the container (3 choices)I pick the flavors (21 choices)

By the product principle there are

3× 21 = 63

different true doubles.

I pick the container (3 choices)

I pick the flavors (21 choices)By the product principle there are

3× 21 = 63

For the first part of the problem(number of two-scoop orders,period), we can break it down intotwo disjoint cases.

I we have a true double

(63 orders)

I both scoops have the same flavor

(21 orders)

The first case is already covered (63orders).

I we have a true double (63 orders)I both scoops have the same flavor

(21 orders)

I we have a true double (63 orders)I both scoops have the same flavor

(21 orders)

For the second case, we need to pickI a container (3 choices)I a single flavor (7 choices)

so there are 3× 7 = 21 two-scoop ofa single flavor.

For the second case, we need to pickI a container (3 choices)I a single flavor (7 choices)

so there are 3× 7 = 21 two-scoop ofa single flavor.

Combining the two cases with thesum principle, there are a total of

63 + 21 = 84

two-scoop ice cream orders.

Next time: Counting poker hands.

Section 3.1.5: Permutations and Combinations · Combination. 4 Example (1) ... This is a...

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