7.2 Pascal’s Triangle and
Combinations
4/10/2013
In today’s lesson we’re learning…
how to find the possible number of combinations
given a situation and how it relates to Pascal’s
triangle.
Factorial !
Definition: The product of an integer and all the
integers below it.
How to
0! = 1
Calculate: 1! = 1
2! = 2•1 =2
3! = 3•2•1 = 6
4!= 4•3•2•1 = 24
Combination
Definition: is a way of selecting several things out
of a larger group, where order does not
matter.
How it is nCk
written:
read as “n Choose k”.
That means that you have n number of
selections and you’re choosing k
amount.
nCk is the number of possible
combinations from that choice.
Example: Ice cream
There are 4 flavors of ice cream you can choose from
and you get to pick 2. How many 2-flavor
combinations can you have?
List of possible combinations:
4 flavors of ice cream
RV
Rocky Road
RM
Vanilla
RS
VM
Mint Chip
VS
Strawberry
MS
There are 6 combinations.
Luckily, there’s a formula you can use instead of making a
list!!! Cool huh?
nCk
nCk
Formula
=
𝑛!
𝑛−𝑘 !𝑘!
For the ice cream example: 4C2 “4 choose 2” since
there are 4 flavors and you get to choose 2.
4C2
=
4!
4−2 !2!
=
4∙3∙2∙1
2∙1∙2∙1
=6
Find the number of combinations:
nCk
=
𝑛!
𝑛−𝑘 !𝑘!
1. 6C2
6C2
=
6!
6−2 !2!
=
6∙5∙4∙3∙2∙1
4∙3∙2∙1∙2∙1
= 15
2. 7C4
7C4
=
7!
7−4 !4!
=
7∙6∙5∙4∙3∙2∙1
3∙2∙1∙4∙3∙2∙1
= 35
So how does this relate to Pascal’s Triangle?
Note: The numbers in the Pascal’s Triangle represents nCk
Now let’s do some word problems!
3 types of problems and what to do.
1. “exactly” – multiply each group
2. “at least” – add each group.
3. “at most” – add each group.
A restaurant gives options of 6 vegetables and 4 meats be ordered in
an omelet. Suppose you want exactly 2 vegetables and 3 meats in
your omelet. How different omelets can you order?
6C2 for
veggies
4C3 for meat
“exactly” multiply each group.
6C2• 4C3 = 15 • 4 = 60
6C2
4C3
You are going to buy a bouquet of flowers. The florist has 18 different
types of flowers. You want exactly 3 types of flowers. How many
different combinations of flowers can you use in your bouquet?
What we have is this: 18C3
Since our Pascal’s Triangle is not big enough to show
the 18th row, let’s use the Combination formula.
nCk
=
18C3
=
𝑛!
𝑛−𝑘 !𝑘!
18!
18−3 !3!
18 ∙ 17 ∙ 16 ∙ 15 … .3 ∙ 2 ∙ 1
=
15 ∙ 14 ∙ 13. . 3 ∙ 2 ∙ 1 ∙ 3 ∙ 2 ∙ 1
=
18∙17∙16
6
= 816
During the school year, the basketball team is scheduled to play 12
home games. You want to attend at least 9 of the games. How
many different combinations of games can you attend?
At least 9 games means you can attend 9, 10, 11, 12.
So ADD all the possibilities!
12C9 + 12C10 + 12C11 + 12C12
220 + 66 + 12 + 1 = 299
You only like 6 songs on the latest Arcade Fire album. If you want to
purchase at most 4 songs with the credit you have on iTunes, how
many different combinations can you buy?
At most 4 songs means you can buy 0, 1, 2, 3 or 4.
So ADD all the possibilities!
6C0 + 6C1 + 6C2 + 6C3+ 6C4
1 + 6 + 15 + 20 +15 = 57
Homework
WS 7.2
Skip #s 8, 9, 12 and 14.
What does a clock do when it
gets hungry???
It goes back four seconds!!!