1. An introduction to Combinations
Slideshow 61, Mathematics
Mr Richard Sasaki, Room 307

2. 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)

3. 1, 2 1, 3 2, 3 1, 4 2, 4 3, 4 Combinations What is a combination?
A combination is an arrangement of objects (numbers, letters, words etc…) where order doesn’t matter.
For example: 1, 2 is the same as
2, 1
Let’s list the possible ways we can pick two numbers from 1, 2, 3 and 4 𝑛=4, 𝑟=2 where repetition is not okay.
1, 2
We get 6 results.
It isn’t obvious how to make a formula is it?
1, 3
2, 3
1, 4
2, 4
3, 4

4. Combinations without repetition
So we’re picking 2 from 4 𝑛=4, 𝑟=2 with no order and no repetition.
If order did matter we’d also get…
1, 2
2, 1
1, 3
2, 3
3, 1
3, 2
1, 4
2, 4
3, 4
4, 1
4, 2
4, 3
C: We get 6 results.
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 (𝑟=2).

5. Combinations without repetition
How many ways are there of ordering a trio (𝑟=3)?
6…right?
How many ways are there to order r items? r!
Formula for permutations without repetition
Formula for combinations without repetition
𝑛! 𝑛−𝑟 !
𝑛! 𝑛−𝑟 !
nPr =
nCr = 𝑟!
Again, all we have to do is simple substitution.

6. Combinations without repetition
Let’s try an example.
Example
There 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=5
𝑟=3
𝑛! 𝑟! 𝑛−𝑟 !
= 5! 3! 5−3 !
= 5! 3!∙2!
= ∙2
5C3 =
= 10 combinations

7. 6 6
120 56 1 91 23

8. 455 210 495 84 715 Because 𝑛! 𝑟! 𝑛−𝑟 ! = 𝑛! 𝑟! =1 when 𝑛=𝑟.
120∙119∙118 6

9. 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…
And we pick three scoops. (Repetition is okay.)
𝑛=5, 𝑟=3
Banana
Chocolate
Lime
Strawberry
Vanilla
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.

10. Combinations with repetition
Banana
Chocolate
Lime
Strawberry
Vanilla
O means choose and → means pass. (𝑛=5, 𝑟=3)
If we want chocolate, lime and vanilla… → O → O → → O
If we want a banana and two vanilla… O → → → → O O
If we want three strawberry… → → → O O O →
The point of this is we always do 7 commands.
In the general case for 𝑛 and 𝑟 we’d always do 𝑟+(𝑛−1) commands (we picked 3 (O) and moved 4 (→)).
So we do 𝑟+(𝑛−1) commands and pick 𝑟 scoops (instead of 𝑛 choices and 𝑟 being picked.

11. Combinations with repetition
Lastly, we insert 𝑛+𝑟−1and 𝑟 scoops into our combinations formula instead of 𝑛 and 𝑟.
So instead of…
𝑛! 𝑟! 𝑛−𝑟 !
nCr =
We get…
(𝑛+𝑟−1)! 𝑟! (𝑛+𝑟−1)−𝑟 !
= (𝑛+𝑟−1)! 𝑟! 𝑛−1 !
n+r-1Cr =
∴n+r−1Cr = (𝑛+𝑟−1)! 𝑟! 𝑛−1 !

12. Combinations with repetition
Let’s try an example.
Example
There 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=5
𝑟=3
(𝑛+𝑟−1)! 𝑟! 𝑛−1 !
= (5+3−1)! 3! 5−1 !
= 7! 3!∙4!
= ∙24
5+3-1C3 =
= 35

13. 10 70 10
231
210

14. 35 56
1365
2002 35
a. 𝑛=5, 𝑟=3 b. 𝑛=12, 𝑟=4
c. 𝑛=10, 𝑟=5
1820
a. 𝑛=7, 𝑟=3 b. 𝑛=15, 𝑟=4
c. 𝑛=14, 𝑟=5

15. 125
720
210
126

16. 125
210
792
He must have eaten at least two of the same type.
210
360