Transcript Document

F. How Many Ways can You Organize?
Math 30: Pre-Calculus
PC30.12
Demonstrate understanding of permutations,
including the fundamental counting principle.
PC30.13
PC30.13 Demonstrate understanding of
combinations of elements, including the application
to the binomial theorem.
Key Terms:
1. Permutations
 PC30.12
 Demonstrate understanding of permutations, including the
fundamental counting principle.
1. Permutations
 Counting Methods are used to determine the number of
members from a set as well as the outcome of an event.
 There are methods such as tables, lists or tree diagrams that
allow you to visually see all the outcomes.
 Another methods for determining the number of outcomes is
called the Fundamental Counting Principle.
 The fundamental counting principle states that if one task can
be performed in a ways and second task can be performed in
b ways, then the two tasks can be performed in (a)(b) ways.
Example 1
 In the last example when we arranged the 5 students in the
middle we end up with (5)(4)(3)(2)(1) ways.
 This can be abbreviated as 5! and is read as 5 factorial.
 Therefore, 5!= (5)(4)(3)(2)(1)
 Seven different objects can arranged 7! Ways
 If there are 7 members of a student council how many ways
can they select the chair, secretary and treasurer?
Example 2
 With permutations we said order of different objects is
important. Well what is some of the objects in the set are
identical.
 Consider the word WEED and all the possible 4 letter
arrangements.
 If all the letters were different the number of outcomes
would be 4!=24
 There are however 2 identical letters. If they were different
we would arrange them 2!=2 ways.
 So the number of arrangements of the word WEED is
 For permutations with repeating objects, a set of n objects
with “a” of one kind that are identical, “b” of a second kind
that are identical, and “c” of a third kind that are identical,
and so one, can be arranged in
Example 3
Example 4
 To solve some problems you must count the different
arrangements in all the cases that together cover all the
possibilities.
 Calculate the number of arrangements for each case and then
add that values for all cases to obtain the total number of
arrangements.
 Whenever you encounter a situation with constraints or
restriction, always address the choices for the restricted
positions first.
 For example, you may need to determine the number of
arrangements of 4 girls and 3 boys in a row of 7 seats if the
end of the rows must be either both male or both female.
Example 5
Key Ideas
p. 526
Practice
 Ex. 11.1 (p.524) #1,2-8 odds in each, 9-18 evens, 22
#5-8 odds in each, 9, 8-26 evens
2. Combinations
 PC30.13
 PC30.13 Demonstrate understanding of combinations of
elements, including the application to the binomial theorem.
2. Combinations
 A combination is a selection of a group of objects taken from
a larger group.
 The kinds of objects selected is important but NOT the
order in which they are selected.
 There are a few ways to find the possible number of
combinations
 On is to use reasoning. Use the fundamental counting
principle and divide by the number of ways that the objects
can be arranged among themselves.
 For example, calculate the number of combinations of 3
digits made from 1-5 without repetition.
 There are 60 ways to arrange 3 items form 5
 However, 3 digits can be arranged 3! Ways among
themselves and in a combination there are considered the
same selection.
 So
 So number of ways of choosing 3 digits from five digits is:
 The number of combinations of “n” items taken “r” at a time
is equivilent to teh number of combinations of n items taken
n-r at a time; that is nCr=nCn-r
 Proof:
 To solve some problems, count the different combinations in
cases that together cover all the possibilities.
 Calculate the number of combinations for each case and then
add the values for all cases to obtain the total number of
combinations.
Example 1
Example 2
Example 3
 When answering questions it is important to know if you are
dealing with a permutation or a combination.
 Remember in Permutations the order of the objects is
important.
 IN the combinations the type of objects is important but
NOT the order in which they are selected.
Key Ideas:
p.533
Practice
 Ex. 11.2 (p.534) #1-6 odds in each, 7-13, 14-20 evens
#4-6 odds in each, 7-13, 14-24 evens
3. The Binomial Theorem
 PC30.13
 PC30.13 Demonstrate understanding of combinations of
elements, including the application to the binomial theorem.
3. The Binomial Theorem
 If you expand a power of a binomial expression, such as
(x+y)4 you get a series of terms
 There are many patterns in the expression of (x+y)4
Example 1
 For coefficients you can use Pascal’s triangle instead of
Combinations.
 Important Properties of the binomial expansion (x+y)n
include:
 Write binomial expansions in descending order of exponent
of the first term in the binomial
 The number of objects, “k”, selected in the combination nCk
can be taken to match the number of factors of the second
variable. That is, it is the same as the exponent on the second
variable.
 The sum of the exponents in any term of the expansion is
“n”.
 The General Term tk+1 has the form:
n-k(y)k
C
(x)
n k
Example 2
Key Ideas:
p.541
Practice
 Ex. 11.3 (p.542) #1-7 odds in each, 8-9, 11-19 odds
#2-3, 5-7 odds in each, 9-25 odds