1. Permutations & Combinations
How do you count arrangements?

2. When arranging objects…
The fundamental counting principle gives you the
number of ways a task can occur given a series of
events.
Suppose you have 5 students
going to the movies: Adam, Brett,
Candice, David and Eva. They are
going to sit in 5 consecutive seats
in one row at the theatre.
To fill the first seat, any of the 5 students can
choose to sit in that seat. After the first student is
seated, any of the 4 remaining students can choose
to be seated in the next seat.

3. Fundamental Counting Principle
We continue in this manner until
we get to the last seat which is
left up to the one remaining
student.
The Fundamental Counting Principle says that
we would multiply the number of ways you can
fill each seat (an event) to get the total number
of orderings.
The solution: 5*4*3*2*1 = 120 ways you can seat
the 5 students in the 5 chairs at the theatre.

4. Let’s count license plates!
Here’s another example using FTC. Suppose you
want to find the number of possible license plates
for cars in North Carolina. The first three positions
on the license plate are for letters and the last four
positions are for digits. All 26 letters of the
alphabet may be used and all ten digits, 0 – 9.
Letters and digits may be repeated.

5. That’s a whole lotta plates!
FTC tells us to multiply the number of ways you can fill each slot
on the license plate. Each of the first 3 slots can be filled 26
different ways by each of the 26 letters of the alphabet. That
means there are 26*26*26 = 17,576 arrangements of the 3
letters alone.
Each of the next four slots can be filled by one of the ten digits,
0, 1, 2, 3, 4, 5, 6, 7, 8, 9. So the ten digits have 10*10*10*10 =
10,000 different arrangements, ranging from 0000 to 9999.
The total number of plates? Why just multiply together these two
numbers: 175,760,000 license plates.

6. Factorials
 Before we jump into permutations & combinations,
you need to understand factorials.
 A factorial is mathematical calculation, like square
root or addition, and is represented with a ! mark.
 n! is the product of the numbers from n down to 1
in the decreasing sequence. It is written as n! =
n*(n-1)*(n-2)*….*2*1.
 So 5! is 5*4*3*2*1 or 120 and 8! = 8*7*6*5*4*3*2*1.
 We will use factorials in permutations &
combinations.

7. Permutations vs Combinations
 Permutations are written as nPx where n is the
number of total choices possible and x is the
number of choices that will be used. This is
calculated as nPx = n!
(n–x)!
 Permutations represent the number of ways we
can choose x objects from n possibilities where the
order of selection matters.
 Combinations represent the number of ways we
can choose x objects from n possibilities where the
order of selection does not matter. This is
calculated as nCx = n!___
x!(n-x)!

8. Using the calculator for !
 Your calculator has built-in functions for
permutations & combinations & factorials.
 When you use nPx or nCx, you have to
enter the n value into your calculator first,
then go to Math  ProbnPx to enter the
function. Type in the value for x last and hit
Enter.
 10 P7 = 10! = 10! = 720
(10-3)! 7!

9. Combination example
 Suppose you are going out to dinner with friends.
The restaurant advertises a 2-for $20 special
where you may choose from one of 5 appetizers, 2
entrees from 10 possible and one dessert from 4
possibilities. You and your friend want to get
different entrees. How many different ways can you
choose the two entrees?
 Whichever of you that will choose first will have 10
selections to choose from and the other person will
have 9 selections to choose from. This is
calculated using permutations.
 10 C2 = 10! = 10! = 45 meal combinations
2!(10-2)! 2! 8!

10. Permutation example
 Permutations are like combinations but here the
order is important. Suppose you have 10 students
running in a race and the top 3 winner receive
medals for 1st, 2nd and 3rd places – a gold, a silver
and a bronze.
 You calculate the number of ways this race can be
won by 10 C3 = 10! = 10*9*8 = 720 ways
(10-3)!

11. Summary
 Fundamental  Order DOES
counting principle matter. Repeats
ARE allowed.
 Permutations  Order DOES
matter. Repeats
are NOT allowed.
 Order does NOT
 Combinations
matter. Repeats
are NOT allowed.