AN INTRODUCTION TO COMBINATIONS Slideshow 61, Mathematics Mr Richard Sasaki, Room 307.
Transcript of AN INTRODUCTION TO COMBINATIONS Slideshow 61, Mathematics Mr Richard Sasaki, Room 307.
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