1. 8.1 Notes.notebook December 18, 2012
Unit 8: Permutations, Combinations and
the Binomial Theorem
This unit deals with logical reasoning and formulas that
can be used for arrangements of objects such as letters, digits,
people, etc. This unit is useful for many post secondary math
courses that are NOT just Calculus centred.
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2. 8.1 Notes.notebook December 18, 2012
8.1 Fundamental Counting Principle
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6. 8.1 Notes.notebook December 18, 2012
Work with factorials to solve (without a calculator!):
a)
b)
Work with factorials to simplify:
a)
b)
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7. 8.1 Notes.notebook December 18, 2012
The fundamental counting principle can be used in questions
that take MORE than 1 step. These are often questions that
involve the word "OR" and are called CASE questions.
To find the TOTAL number of arrangements, you find the
answer 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 the
numbers 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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8. 8.1 Notes.notebook December 18, 2012
More case question examples:
1) How many 3 digit numbers less than 460 can be
made from the digits 0­9. (No repeats0
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9. 8.1 Notes.notebook December 18, 2012
2) Using 1,2,3,4,8,9 (No repeats) how many 3 digit numbers
can be made that are greater than 430.
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10. 8.1 Notes.notebook December 18, 2012
Repetitions:
How many different arrangements are possible using the
letters of the word CAT?
(We do NOT have to ask if repeats are allowed since we
can only use each letter ONCE)
How many different arrangements are possible using the
letters of the word BOO?
What has changed from the first question?
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11. 8.1 Notes.notebook December 18, 2012
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 of
each 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:
ways
Ex) How many different arrangements are possible of the letters
of the word MISSISSIPPI.
Ex) How many different 5­digit number can be made by arranging
the digis of 17 171?
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12. 8.1 Notes.notebook December 18, 2012
More examples:
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13. 8.1 Notes.notebook December 18, 2012
HOMEWORK: Assignment is on next page.
Also, supplement assignment from Smart Board
for tomorrow.
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