AN INTRODUCTION TO COMBINATIONS

Slideshow 61, Mathematics

Mr Richard Sasaki, Room 307

OBJECTIVES

• Understand the meaning of a combination

• Recall how to calculate combinations without repetition (without replacement)

• Learn how to calculate combinations with repetition (with replacement)

COMBINATIONS

What is a combination?

A combination is an arrangement of objects (numbers, letters, words etc…) where order doesn’t matter.Let’s list the possible ways we can pick two numbers from 1, 2, 3 and 4 where repetition is not okay.

1, 21, 31, 4

We get 6 results.

It isn’t obvious how to make a formula is it?

2, 32, 4 3, 4

For example: 1, 2 is the same as .2, 1

COMBINATIONS WITHOUT REPETITION

So we’re picking 2 from 4 with no order and no repetition.

1, 21, 31, 4

C: We get 6 results.

2, 32, 4 3, 4

If order did matter we’d also get…

2, 13, 14, 1

3, 24, 2 4, 3

P: We get 12 results.

Why is the number of combinations half of the number of permutations?Because permutations involve order and there are two ways of ordering a pair .

COMBINATIONS WITHOUT REPETITION

How many ways are there of ordering a trio ?6…right?

How many ways are there to order r items?r!Formula for

permutations without repetition

nPr =𝑛 !

(𝑛−𝑟 ) !

Formula for combinations without

repetitionnCr =

𝑛 !(𝑛−𝑟 ) !𝑟 !

Again, all we have to do is simple substitution.

COMBINATIONS WITHOUT REPETITION

Let’s try an example.ExampleThere are 5 bears in the park and food is given to them. Only 3 bears are successful in getting food. How many combinations of the bears getting food are there?

𝑛=5𝑟=3

5C3 =𝑛 !

𝑟 ! (𝑛−𝑟 ) != = = = combinations

6

120

91

23

6

56

1

Because when .

210

84

715

455

495

120 ∙119 ∙1186

COMBINATIONS WITH REPETITION

This is our final challenge for Chapter 6. Unfortunately there isn’t any known good way to prove it, just kind of get the result from patterns.

Let’s imagine that we have ice cream in 5 different boxes like so…

Banana Chocolate Lime Strawberry Vanilla

And we pick three scoops. (Repetition is okay.)

The problem is, a robot is taking our order and we need to give him our order in ‘O’ and ‘’ symbols where O means choose and means pass. All orders start at banana…and end at vanilla.

𝑛=5 ,𝑟=3

COMBINATIONS WITH REPETITION

Banana Chocolate Lime Strawberry Vanilla

O means choose and means pass. () If we want chocolate, lime and vanilla…→O→O→→OIf we want a banana and two vanilla… O→→→→OOIf we want three strawberry… →→→OOO→The point of this is we always do 7 commands.In the general case for and we’d always do commands (we picked 3 (O) and moved 4 ()).So we do commands and pick scoops (instead of choices and being picked.

COMBINATIONS WITH REPETITION

Lastly, we insert and scoops into our combinations formula instead of and .So instead of…

nCr =𝑛 !

𝑟 ! (𝑛−𝑟 ) !We get…

n+r-1Cr =(𝑛+𝑟 −1)!

𝑟 ! ((𝑛+𝑟 −1)−𝑟 ) !¿(𝑛+𝑟 −1)!𝑟 ! (𝑛−1 )!

∴n+r −1C r  =(𝑛+𝑟 −1)!𝑟 ! (𝑛−1 ) !

COMBINATIONS WITH REPETITION

Let’s try an example.ExampleThere are 5 bears in the park and 3 portions of food are given to them. Only up to 3 bears are able to get food but it is possible for a bear to eat 1, 2 or 3 portions. What are the combinations that the bears can eat food?𝑛=5𝑟=3

5+3-1C3 =(𝑛+𝑟−1)!𝑟 ! (𝑛−1 )!

= = = =

10

10

70

231

210

56

35

1820

3513652002

a. a. b.

b. c.

a. a. b.

b. c.

125

210

720

126

125

792

360

210

He must have eaten at least two of the same type.

210