Today in Pre-Calculus
• Review Chapter 9 – need a calculator
• Homework
• Go over Chapter 8 worksheet questions
Combinatorics
• An arrangement of objects in a specific order or selecting
all of the objects.
• An arrangement of objects in which order does not
matter.
• Difference between permutations and combinations:
– Combinations: grouping of objects
– Permutation: putting objects in specific places or positions, or
selecting all of the objects.
ExampleS
1)There are ten drivers in a race. How many
outcomes of first, second, and third place are
possible?
2)In a study hall of 20 students, the teacher can
send only 6 to the library. How many ways can
the teacher send 6 students?
Conditional Probability &
Tree Diagrams
Two identical cookie jars are on a counter. Jar A contains
2 chocolate chip and 2 peanut butter cookies, while jar B
contains 1 chocolate chip cookie. Selecting a cookie at
random, what is the probability that it is a chocolate chip
cookie?
Conditional Probability
Notation: P(A|B) probability of A given B
P(chocolate chip|jar A)=
P(chocolate chip|jar B)=
P(A|B)= P( A and B)
P( B)
P(jar A|chocolate chip) =
Binomial Distribution
Let p be the probability of event A and q be the probability of
event A not occurring given n trials. Then the probability A
occurs r times is
rqn-r
C
p
n n-r
Ex: We roll a fair die four times. What is the probability that
we roll:
a)All 3’s
b) no 3’s
c) Exactly two 3’s
Binomial Theorem
(a + b)n = nC0an + nC1an-1b + nC2an-2b2 + … +
2bn-2 + C abn-1 + C bn
C
a
n n-2
n n-1
n n
Example: (2x2 – 3y)4 =
Find the x6y5 term in the expansion of (x + 3y)11
Sequences
• Arithmetic Sequence: a sequence in which there is a
common difference between every pair of successive terms.
Example: 5,8,11,14
General formula: an = a1 + (n-1)d
• Geometric: a sequence in which there is a common ratio (or
quotient) between every pair of successive terms.
Example: 1 , 1 , 1 , 1 ,....
2 4 8 16
General formula: an = a1r(n–1)
Explicitly Defined Sequence
• A formula is given for any term in the sequence
Example: ak = 2k - 5
Find the 20th term for the sequence
Write the explicit rule for the sequence 55, 49, 43, …
Write the explicit rule for the sequence 5, 10, 20, …
Series
•
•
•
•
Series: the sum
of the terms of a sequence {a1, a2, …,an}
n
Written as:  ak
k 1
Read as “the sum of ak from k = 1 to n”.
k is the index of summation
5
 2k 
k 1
6
k
2
 
k 3
formulas
n
n
n
arithmetic :  ak   a1  an    2a1  (n  1)d 
2
2
k 1
100
Example :  k 
k 1
n
Geometric :  ak 
k 1
a1 1  r n 
1 r
1
Example :  28  
2
k 1
10
k 1

Example
Write the sum of the following series using summation
notation:
Example 1: 13 + 17 + 21 + … + 49
Example 2: 1 + 8 + 27 + … + (n+1)3
Example 3: 3, 6, 12, …, 12,288
Infinite Series
An infinite series can either:
1) Converge – if, as n increases, the series sum approaches a
value (S)
2) Diverge – if as n increases, the series sum does NOT
approach a value.
¥
k-1
a
ir
converges iff r < 1
å
k=1
¥
If it does converge, the å a ir
k=1
k-1
a1
=
1- r

 3
Does this series converge ? If so, give the sum.  7   
5
k 1 
k 1
Homework
•
•
•
•
Pg 708: 2,4,11,19,21,22
Pg 715: 13,15,19,21
Pg 728: 31,33, 45-50
Pg 787: 55-58, 63,70,77-81odd,83,84,