1. Permutations and Combinations
Determining Probabilities in Carbon Nanostructures

2. Permutations Notice, ORDER MATTERS!
A Permutation is an arrangement of items in a particular order.
Notice, ORDER MATTERS!
To find the number of Permutations of n items, we can use the Fundamental Counting Principle or factorial notation. 2

3. Permutations
To find the number of Permutations of n items chosen r at a time, you can use the formula 3

4. Permutations Practice:
A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated?
Answer Now 4

5. Permutations Practice:
From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled? 5

6. Combinations ORDER DOES NOT MATTER!
A Combination is an arrangement of items in which order does not matter.
ORDER DOES NOT MATTER!
Since the order does not matter in combinations, there are fewer combinations than permutations.  The combinations are a "subset" of the permutations. 6

7. Combinations
To find the number of Combinations of n items chosen r at a time, you can use the formula 7

8. Combinations
To find the number of Combinations of n items chosen r at a time, you can use the formula 8

9. Combinations
Practice:
A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions?
Answer Now 9

10. Combinations
A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions?
Practice: 10

11. Determining Probabilities
Example:
A Class has 12 girls and 10 boys
How many ways to choose 3 girls and 2 boys?
What is the probability that 5 students chosen at random are 3 girls and 2 boys?

12. First Use combinations, since order is not important
Girls x Boys
C(n,r) C(n,r)
C(12,3) C(10,2)
Probability that 5
Students chosen from
Class of 22 are 3 girls
And 2 boys is P

13. Closer Look at some of the Allotropes of Carbon

14. Allotropes of Carbon
An allotrope is a variant of a substance consisting of only one type of atom.
Diamond
graphite (many layers of graphene)
Fullerenes (buckyballs)
Carbon Nanotubes (CNTs)

15. Diamond
Diamond has a three-dimensional network structure in which each carbon is singly-bonded to four others with sp3 hybridization.
Diamond is a covalent network solid
each carbon covalently bonded to 4 others.
Diamonds are the hardest substance known. 2

16. Graphene
Graphene is a form of carbon. Not only the thinnest ever but also the strongest.
Graphene can be of any size, from few carbon atoms to billions of them.

17. Graphite
Graphite has a layered structure, where graphene makes up each layer. The edges are very reactive! 2

18. Carbon Nanotubes (CNTs)
Carbon nanotubes are formed by a layer of hexagonally-arranged carbon atoms rolled into a cylinder - usually have half buckyballs on one or both ends
The Edges of Carbon Nanotubes are also very reactive!
If functional groups are to be attached, they will attach only on those carbon atoms!
Carbon nanotube length can be a million times greater than its width

19. Fullerenes (Buckyballs)
The fullerenes are a family of molecules with a closed cage of carbon atoms arranged in pentagons and hexagons. C60 is an example, but there could be other types as well 2

20. Now you will determine the probability of attaching functional groups to the Carbon Nanostructures. Remember to use Combinations and Permutations when necessary.