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FACTORIALS, FUNDAMENTAL COUNTING
PRINCIPLE, COMBINATIONS AND
PERMUTATIONS
Last updated on 12/5/2015 3:27:00 PM
OVERVIEW
1. Factorials
What is a
factorial?
A factorial is the product of a positive integer and the integers below it. NOTE: Product means that we are multiplying. The symbol that is used for factorials is an exclamation point (!). For example,
๐๐! = ๐๐ ร ๐๐ ร ๐๐ ร ๐๐ ร ๐๐^1 ๐๐๐๐! = ๐๐๐๐ ร ๐๐ ร ๐๐ ร ๐๐ ร ๐๐ ร ๐๐ ร ๐๐ ร ๐๐ ร ๐๐ ร ๐๐ Keep in mind! The exception to the rule is zero factorial which is equal to one. ๐๐! = ๐๐
How is it used? In terms of the order of operations (PEMDAS), consider calculating
factorials between parenthesis and exponents e.g.
- Parenthesis 2. Factorials
- Exponents
- Multiplication
- Division
- Addition
- Subtraction A couple of examples on simplifying equations with factorials are below: Example 1:
๐๐! ๐๐!
๐๐ ร ๐๐ ร ๐๐ ร ๐๐
๐๐ ร ๐๐ ร ๐๐ ร ๐๐ ร ๐๐ ร ๐๐ ร ๐๐ ร ๐๐
๐๐ ร ๐๐ ร ๐๐ ร ๐๐
Example 2:
3! ร 2! = (๐๐ ร ๐๐ ร ๐๐) ร (๐๐ ร ๐๐) = 6 ร 2 = 12
(^1) 5! Is read as โ5 factorialโ. 10! Is read as โ10 factorial.โ
FACTORIALS, FUNDAMENTAL COUNTING
PRINCIPLE, COMBINATIONS AND
PERMUTATIONS
Last updated on 12/5/2015 3:27:00 PM
Where are
additional
references?
Understanding factorials is the foundation for calculating combinations and permutations. The Jing video http://screencast.com/t/g1W1KhDaAQ includes tips for searching for information for factorials online.
Calculating 5! (Visually Speaking)
FACTORIALS, FUNDAMENTAL COUNTING PRINCIPLE,
COMBINATIONS AND PERMUTATIONS
Last updated on 12/5/2015 3:27:00 PM
Calculating the Fundamental Counting Principle Using a Tree Diagram (Visually Speaking)
The answer to the example (different meals) can also be calculated using a tree diagram. Although a tree diagram helps to see each combination, from an efficiency standpoint, it takes more time. As a result, in preparing for the CASA and CORE exams, understanding the fundamental counting principle is key. A tree diagram (if time permits) can be used to check the answer for a problem.
FACTORIALS, FUNDAMENTAL COUNTING
PRINCIPLE, COMBINATIONS AND
PERMUTATIONS
Last updated on 12/5/2015 3:27:00 PM
3. Combinations
What is a
combination?
A combination is the act of arranging elements into specified groups without regard to order.
Think of a combination as the different ways that you can choose/pick X objects out of Y objects.
In the combination equation, n is the total number of objects and r is the number that you are choosing/picking/arranging each time. SIDE NOTE: r is less than or equal to n (๐๐ โค ๐๐). r cannot be greater than n. For example, if you have a blue marble, a red marble, and a green marble, how many different ways can we pick 2 marbles? In this case, n is equal to 3 (the total number of marbles) and r is equal to 2 (the number of marbles we are picking). As a result,
๐๐ ร ๐๐ ร ๐๐
๐๐ ๐๐ ร ๐๐(๐๐!)
When we pick 2 marbles, the possible combinations are Blue , Green Green , Red Blue , Red NOTE: Picking a blue marble first and a green marble second would be the same as picking a green marble first and a blue marble second. Order does not matter with combinations. With combinations, youโre looking at the final result.
How is it used? If you^ are reading a word problem and you are trying to determine if
it is a combination problem or not, ask yourself the question, โ Does order matter? โ If order matters, the problem is not a combination.
Where are
additional
references?
You can find additional practice problems for combinations by using a search engine to find โpractice problems combinationโ. The resulting searches using Google, Dogpile, and Mamma are below:
FACTORIALS, FUNDAMENTAL COUNTING
PRINCIPLE, COMBINATIONS AND
PERMUTATIONS
Last updated on 12/5/2015 3:27:00 PM
Where are
additional
references?
You can find additional practice problems for permutations by using a search engine to find โpractice problems permutationโ. The resulting searches using Google, Dogpile, and Mamma are below:
5. Fundamental Counting Principle versus Combinations versus Permutations
When trying to decide if you should use the fundamental counting principle versus a combination versus a permutation, ask the following questions:
- Am I looking at how many different ways to combine 2 or more groups? If so, use the fundamental counting principle.
- Does order matter? If so, use permutations.
- Am I grouping objects e.g. how many different ways to choose 3 out of 5? If so, use combinations.
