1. Permutations and Combinations
Fundamental Counting Principles
EQ: What is the difference between a permutation and a combination?

2. Fundamental Counting Principle
In a sequence of n events in which the first one has k1 possibilities, the second has k2 possibilities and the 3rd has k3 and so on, the total number of possibilities of the sequence will be
k1 · k2 · k3 ·… ·kn

3. Example 1 A buffet menu consists of the following selections
entrée: chicken, beef, pork, pasta
vegetable: corn, string beans, peas
potato: baked, french fries, scalloped
Find the total number of possible dinner combinations if you can only select one of each item.
4·3·3 = 36

4. Example 2:
A photographer has 5 photographs that she can mount on a page in her portfolio. How many different ways can she arrange her photographs?
= 120 5 4 3 2 1

5. Example 3:
The digits 0, 1, 2, 3, and 4 are to be used in a four-digit ID card. How many different cards are possible
if repetitions are permitted?
if repetitions are not permitted? 5 4 3 2 5 5 5 5 b)
= 120 a)
= 54 = 625

6. Example 4:
Using the digits 1, 2, 3, and 5, how many 4 digit numbers can be formed if
The 1st digit must be a 1 and you can repeat digits?
The 1st digit must be a 1 and you can’t repeat digits?
The number must be divisible by 2 and you can repeat digits?
The 1st number must be prime and you can’t repeat digits.

7. Example 4:
Using the digits 1, 2, 3, and 5, how many 4 digit numbers can be formed if 1 4 4 4 1 3 2 1 a)
= 43 = 64 b)
= 6 4 4 4 1 3 3 2 1 c)
= 64 d)
= 18

8. Example 5:
A father, mother, 2 boys, and 3 girls are asked to line up for a photograph. Determine the number of ways they can line up if…
there are no restrictions on how they stand.
the parents must stand together.
the parents do not stand together.
all females must stand together.

9. Example 5: 7 6 5 4 3 2 1 a) ______ ______ ______ 6 1
= 5040 6 1
b) ______ ______ ______
= 1440
(2X1) 5 4 3 2
c) part a – part b
= = 3600
d) ______ ______ ______
= 576
(4x3x2x1) 4 3 2 1

10. PERMUTATIONS An arrangement of ‘n’ objects in a specific order.
3 basic types:
The permutation of all ‘n’ objects is n!
n! = n(n-1)(n-2)….·1

11. Example: Permutation Rule 1
How many ways can 9 trophies be displayed on a shelf?
9! = 362,880

12. Permutation Rule 2
Use when you need to arrange only ‘r’ of the original ‘n’ objects.

13. Example: Permutation Rule 2
10 ladies are up for Miss Universe. How many different ways can they be ranked?
10! = 3,628,800
Of the ten ladies, how many ways can they pick the top 3 contestants?
= 720

14. Permutation Rule 3
To be used when you need to arrange ‘n’ objects of which k1, k2,…etc are alike.

15. Example: Permutation Rule 3
Find the number of ways you can arrange the letters of the following names:
Tyler
5! = 120
Shannon
7!/3! = 840
Jennifer
8!/(2!2!) = 10080

16. Summarizing Video on Permutations

17. COMBINATIONS
A selection of ‘r’ objects from ‘n’ objects without regard for order.

18. Example 1: Combinations
A University catalog lists 15 graduate courses in physics. Calculate the number of ways in which a student can select 4 courses to take.
= 1365

19. Example 2: Combinations
A travel brochure lists 10 museums in the city of London. In how many ways can a tourist visit 4 museums if
The order in which the museums are visited does not matter
The order in which the museums are visited does matter?

20. Example 2: Combinations - Answers
Since the order does not matter, this part of the problem is a combination
Since the order does matter, this part of the problem is a permutation

21. Example 3: Combinations
A student committee must consist of 3 juniors and 4 seniors. If seven juniors and eight seniors are willing to serve on the committee, in how many different ways can the committee be formed?
= 35 · 70 = 2450

22. Example 4: Combinations
Tyler wants chocolate ice cream in a dish with 2 different toppings. How many different combinations are possible?
His Pappy wants 3 different toppings on any one flavor of ice cream. How many different combinations are possible?
Flavors
Toppings
Vanilla
Hot fudge
Chocolate
Sprinkles
Peanut butter
Peanuts
M & M’s

23. Summarizing Video on Combinations