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Combinations. Combinations are very similar to permutations with one key difference: Order does not matter. Combinations. Example 1: You have selected 5 cities to visit on a vacation. How many possible itineraries do you have? (In how many different orders could you visit the 5 cities?

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Combinations

Combinations

Combinations are very similar to permutations with one key difference:

Order does not matter.


Combinations1

Combinations

Example 1: You have selected 5 cities to visit on a vacation. How many possible itineraries do you have? (In how many different orders could you visit the 5 cities?

Example 2: You have a list of 15 cities that you want to visit, but you can only fit 5 into your schedule. How many different sets of 5 cities could you select?

Consider two very similar examples that illustrate this difference.


Combinations2

Combinations

The first example is a permutation problem, because order matters.

The second example is a combination, because we are looking only for a grouping, and order does not matter.


Combinations3

Combinations

Because the problems are similar, it might not surprise you that the methods for solving them are also similar.


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Combinations

Example: You have 5 movies that you want to watch, but you only have time for 2 of them. How many different pairs of movies could you select?

Let’s begin with an even simpler problem.


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Combinations

We could start the problem the same way we would start a permutation problem.

You have 5 options for the first movie. After picking 1 movie, you have 4 options remaining for the second movie.

5*4 = 20, but we need to adjust this.

Select 2 out of 5 movies.


Combinations6

Combinations

Label the movies as A, B, C, D, and E.

Consider the pairs where A is the first movie:

AB

ACADAE

Select 2 out of 5 movies.


Combinations7

Combinations

Label the movies as A, B, C, D, and E.

Add the pairs where B is the first movie:

AB BA

AC BCAD BDAE BE

Select 2 out of 5 movies.


Combinations8

Combinations

Label the movies as A, B, C, D, and E.

Add the pairs where C is the first movie:

AB BA CA

AC BC CBAD BD CDAE BE CE

Select 2 out of 5 movies.


Combinations9

Combinations

Label the movies as A, B, C, D, and E.

Add the pairs where D is the first movie:

AB BA CA DA

AC BC CB DBAD BD CD DCAE BE CE DE

Select 2 out of 5 movies.


Combinations10

Combinations

Label the movies as A, B, C, D, and E.

Add the pairs where E is the first movie:

AB BA CA DA EA

AC BC CB DB EBAD BD CD DC ECAE BE CE DE ED

Select 2 out of 5 movies.


Combinations11

Combinations

Label the movies as A, B, C, D, and E.

Have you spotted the problem?

AB BA CA DA EA

AC BC CB DB EBAD BD CD DC ECAE BE CE DE ED

Select 2 out of 5 movies.


Combinations12

Combinations

Label the movies as A, B, C, D, and E.

Every pair is listed twice.

ABBACADA EA

ACBCCBDBEBADBDCDDCECAE BECEDEED

Select 2 out of 5 movies.


Combinations13

Combinations

Label the movies as A, B, C, D, and E.

We need to divide our answer by 2, to account for the pairs being counted twice.

ABBACADA EA

ACBCCBDBEBADBDCDDCECAE BECEDEED

Select 2 out of 5 movies.


Combinations14

Combinations

Label the movies as A, B, C, D, and E.

(5*4) / 2 = 10 possible pairs of movies.

ABBACADA EA

ACBCCBDBEBADBDCDDCECAE BECEDEED

Select 2 out of 5 movies.


Combinations15

Combinations

Use the labels again for the movies, and begin by listing trios that start with A.

ABC ABD ABE ACB ACD ACE ADB ADC ADE AEB AEC AED

Now say you want to select 3 out of 5 movies.


Combinations16

Combinations

Add the trios that begin with B:

ABC ABD ABE ACB ACD ACE ADB ADC ADE AEB AEC AED BAC BAD BAE BCA BCD BCE BDA BDC BDE BEA BEC BED

Now say you want to select 3 out of 5 movies.


Combinations17

Combinations

Add the trios that begin with C:

ABC ABD ABE ACB ACD ACE ADB ADC ADE AEB AEC AED BAC BAD BAE BCA BCD BCE BDA BDC BDE BEA BEC BED CAB CAD CAE CBA CBD CBE CDA CDB CDE CEA CEB CED

Now say you want to select 3 out of 5 movies.


Combinations18

Combinations

Add the trios that begin with D and E:

ABC ABD ABE ACB ACD ACE ADB ADC ADE AEB AEC AED BAC BAD BAE BCA BCD BCE BDA BDC BDE BEA BEC BED CAB CAD CAE CBA CBD CBE CDA CDB CDE CEA CEB CED DAB DAC DAE DBA DBC DBE DCA DCB DCE DEA DEB DEC EAB EAC EAD EBA EBC EBD ECA ECB ECD EDA EDB EDC

Now say you want to select 3 out of 5 movies.


Combinations19

Combinations

How many times has each trio been counted?

ABC ABD ABE ACB ACD ACE ADB ADC ADE AEB AEC AED BAC BAD BAE BCA BCD BCE BDA BDC BDE BEA BEC BED CAB CAD CAE CBA CBD CBE CDA CDB CDE CEA CEB CED DAB DAC DAE DBA DBC DBE DCA DCB DCE DEA DEB DEC EAB EAC EAD EBA EBC EBD ECA ECB ECD EDA EDB EDC

Now say you want to select 3 out of 5 movies.


Combinations20

Combinations

How many times has each trio been counted?

Not twice, not thrice, but 6 times!

ABC ABD ABE ACB ACD ACE ADB ADC ADE AEB AEC AED BAC BAD BAE BCA BCD BCE BDA BDC BDE BEA BEC BED CAB CAD CAE CBA CBD CBE CDA CDB CDE CEA CEB CED DAB DAC DAE DBA DBC DBE DCA DCB DCE DEA DEB DEC EAB EAC EAD EBA EBC EBD ECA ECB ECD EDA EDB EDC

Now say you want to select 3 out of 5 movies.


Combinations21

Combinations

How many times has each trio been counted?

Look for example at the combination of A, B, and C.

ABC ABD ABE ACB ACD ACE ADB ADC ADE AEB AEC AED BAC BAD BAE BCA BCD BCE BDA BDC BDE BEA BEC BED CAB CAD CAE CBA CBD CBE CDA CDB CDE CEA CEB CED DAB DAC DAE DBA DBC DBE DCA DCB DCE DEA DEB DEC EAB EAC EAD EBA EBC EBD ECA ECB ECD EDA EDB EDC

Now say you want to select 3 out of 5 movies.


Combinations22

Combinations

How many times has each trio been counted?

We want to count this trio once, not 6 times.

ABC ABD ABE ACB ACD ACE ADB ADC ADE AEB AEC AED BAC BAD BAE BCA BCD BCE BDA BDC BDE BEA BEC BED CAB CAD CAE CBA CBD CBE CDA CDB CDE CEA CEB CED DAB DAC DAE DBA DBC DBE DCA DCB DCE DEA DEB DEC EAB EAC EAD EBA EBC EBD ECA ECB ECD EDA EDB EDC

Now say you want to select 3 out of 5 movies.


Combinations23

Combinations

How many times has each trio been counted?

We want to count this trio once, not 6 times. As a result, we must divide by 6.

ABC ABD ABE ACB ACD ACE ADB ADC ADE AEB AEC AED BAC BAD BAE BCA BCD BCE BDA BDC BDE BEA BEC BED CAB CAD CAE CBA CBD CBE CDA CDB CDE CEA CEB CED DAB DAC DAE DBA DBC DBE DCA DCB DCE DEA DEB DEC EAB EAC EAD EBA EBC EBD ECA ECB ECD EDA EDB EDC

Now say you want to select 3 out of 5 movies.


Combinations24

Combinations

How many groups of 3 movies are possible?

(5*4*3) / 6 = 10 possible trios.

ABC ABD ABE ACB ACD ACE ADB ADC ADE AEB AEC AED BAC BAD BAE BCA BCD BCE BDA BDC BDE BEA BEC BED CAB CAD CAE CBA CBD CBE CDA CDB CDE CEA CEB CED DAB DAC DAE DBA DBC DBE DCA DCB DCE DEA DEB DEC EAB EAC EAD EBA EBC EBD ECA ECB ECD EDA EDB EDC

Now say you want to select 3 out of 5 movies.


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Combinations

When we grouped 2 items, we had to divide by 2. When we grouped 3 items, we had to divide by 6. What about for 4 items?

How do you decide the extent of overcounting for these problems?


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Combinations

When we grouped 2 items, we had to divide by 2. When we grouped 3 items, we had to divide by 6. What about for 4 items?

You can answer that by answering this question: In how many ways can you arrange 4 items? That’s how many times each group will be counted, and we need to divide by that factor.

How do you decide the extent of overcounting for these problems?


Combinations27

Combinations

Fortunately, no you do not need to list all possible groups. The number of times a group of 4 items will be counted, for example, is simply the number of permutations for 4 items. We already have a nice formula to use to calculate that number.

Do you need to list every possible group in order to determine how many times each group is counted?


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Combinations

Example: You have a stack of 10 books that you want to read. On your next vacation you decide to take 4 of them with you. In how many ways can you pick 4 books out of the 10?


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Combinations

Example: You have a stack of 10 books that you want to read. On your next vacation you decide to take 4 of them with you. In how many ways can you pick 4 books out of the 10?

You could begin the same as you would with a permutation problem:

P(10, 4) = __10!__ = 10!

(10-4)! 6!


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Combinations

Next we need to consider how many times each group of 4 will have been counted, which is simply 4! times.

We need to divide by that factor to eliminate the overcounting.

Pick 4 out of 10 books.


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Combinations

10!

__6!__ = __10!__

4! 6! 4!

Pick 4 out of 10 books.


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Combinations

__n!__

__(n-m)!__ = ___n!____

m! (n-m)! m!

In general, the number of combinations of m items out of n is:


Combinations33

Combinations

As with permutations, we have some common notations for combinations:

C(n, m)

nCm


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Combinations

Once again, it is common for graphing calculators and computer spreadsheet programs to have a function for combinations.

Graphing calculators typically have a nCr function nearby the nPr function.


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Combinations

Example 1: You have selected 5 cities to visit on a vacation. How many possible itineraries do you have? (In how many different orders could you visit the 5 cities?

Example 2: You have a list of 15 cities that you want to visit, but you can only fit 5 into your schedule. How many different sets of 5 cities could you select?

Go back to our original examples.


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Combinations

Example 1 is a straight forward permutation problem. There are 5!, or 120 different ways to arrange the order of visiting 5 cities.

Go back to our original examples.


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Combinations

Example 2 is a combination problem where we are picking 5 out of 15 cities, and are not concerned with order.

C(15, 5) = __15!__

10! 5!

Go back to our original examples.


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Combinations

  • Enter 15, the total number of items

  • Press MATH

  • Move the cursor to the PRB section

  • Move the cursor down to the nCr function

  • Press ENTER

  • Enter 5, the number of items being chosen

  • Press ENTER

  • You should see an answer of 3,003

On a graphing calculator:


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Combinations

Example: From a list of 5 candidates for student council representative, you are asked to vote for 2. In how many ways can you do this?


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Combinations

You are being asked to pick 2, not rank them, so this is a combination problem.

C(5, 2) = 10

Example: From a list of 5 candidates for student council representative, you are asked to vote for 2. In how many ways can you do this?


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Combinations

Example: From a list of 5 candidates for student council representative, you are asked to vote for 3. In how many ways can you do this?


Combinations42

Combinations

You are being asked to pick 3, not rank them, so this is a combination problem.

C(5, 3) = 10

Example: From a list of 5 candidates for student council representative, you are asked to vote for 3. In how many ways can you do this?


Combinations43

Combinations

You are being asked to pick 3, not rank them, so this is a combination problem.

C(5, 3) = 10 Notice anything interesting?

Example: From a list of 5 candidates for student council representative, you are asked to vote for 3. In how many ways can you do this?


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Combinations

How could this be?

It appears that C(5, 2) = C(5, 3).


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Combinations

Compare the formulas:

__5!__ vs. __5!__

3! 2! 2! 3!

They’re the same thing! But why?

C(5, 2) = C(5, 3)


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Combinations

This is another version of the old glass half full / glass half empty scenario.

C(5, 2) = C(5, 3)


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Combinations

This is another version of the old glass half full / glass half empty scenario.

Choosing which 2 candidates out of 5 to pick is really the same thing as choosing which 3 candidates not to pick.

C(5, 2) = C(5, 3)


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Combinations

In some cases we can combine the technique of combinations with other techniques, such as permutations or the multiplication or addition principles.


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Combinations

Example: At a certain ice cream store, you can make a sundae with either 1 or 2 scoops of ice cream, combined with 1 topping. This store has 20 flavors of ice cream, and 5 different toppings. How many unique sundaes are possible?


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Combinations

First, consider the ice cream.

If you choose 1 scoop of ice cream, there are 20 options (20 flavors).

If you choose 2 scoops, there are (20*20) / 2, or 200 possibilities. (You have 20 options for the first scoop, and 20 again for the second scoop, because you could pick the same flavor for both scoops. We have to divide by 2, because every pair was counted twice.


Combinations51

Combinations

First, consider the ice cream.

Because we have 20 possibilities for 1 scoop, and 200 possibilities for 2 scoops, we end up with 200 + 20, or 220 possibilities for ice cream.


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Combinations

Now consider the topping.

We have 5 options for the topping.


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Combinations

toppings

Now multiply the options.

220*5 = 1100 total possible sundaes

ice cream


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