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8.1 Notes.notebook
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December 18, 2012
May 22:41 PM
Unit 8: Permutations, Combinations and the Binomial Theorem
This unit deals with logical reasoning and formulas thatcan be used for arrangements of objects such as letters, digits,people, etc. This unit is useful for many post secondary mathcourses that are NOT just Calculus centred.
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8.1 Fundamental Counting Principle
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Work with factorials to solve (without a calculator!):
a)
b)
Work with factorials to simplify:
a)
b)
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The fundamental counting principle can be used in questionsthat take MORE than 1 step. These are often questions thatinvolve the word "OR" and are called CASE questions.
To find the TOTAL number of arrangements, you find theanswer for each of the cases, and then ADD them together.This is known as the "Addition Principle of the Fundamental Counting Principle".
Ex) How many numbers are less than 300 if NO repeats are allowed. The problem here is that there is no demand for thenumbers to have only 3 digits. So the answer will be the TOTAL of the 1 digit OR the 2 digit OR the 3 digit numbers.You therefore have 3 cases.
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More case question examples:
1) How many 3 digit numbers less than 460 can bemade from the digits 09. (No repeats0
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2) Using 1,2,3,4,8,9 (No repeats) how many 3 digit numberscan be made that are greater than 430.
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Repetitions:
How many different arrangements are possible using theletters of the word CAT? (We do NOT have to ask if repeats are allowed since wecan only use each letter ONCE)
How many different arrangements are possible using theletters of the word BOO?What has changed from the first question?
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May 23:14 PM
If objects are identical then there will be fewer DIFFERENT arrangements since some of the arrangements will simply switch identical objects.We still use the factorial concept but now divide by the factorials ofeach repeating object to reduce to the correct answer.
Def'n: A set of n objects with a identical, b identical and c identical objects can be arranged in:
Ex) How many different arrangements are possible of the lettersof the word MISSISSIPPI.
ways
Ex) How many different 5digit number can be made by arranging the digis of 17 171?
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More examples:
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HOMEWORK: Assignment is on next page.Also, supplement assignment from Smart Boardfor tomorrow.
