Download A_Counting, Perm, Combinations

Counting,
Permutations, &
Combinations
A counting problem asks
“how many ways” some
event can occur.
Ex. 1: How many three-letter codes are
there using letters A, B, C, and D if no
letter can be repeated?
• One way to solve is to list all
possibilities.
Ex. 2: An experimental psychologist uses
a sequence of two food rewards in an
experiment regarding animal behavior.
The food rewards consists of three
options. How many different sequences
of rewards are there if each option can be
used only once in each sequence?
Next slide
•Another way to solve is a factor
tree where the number of end
branches is your answer.
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Fundamental Counting Principle
• If there are m ways for Event A to
occur and n ways for Event B to
occur, then there are m × n ways for
Events A and B to occur together.
Fundamental Counting Principle
Suppose you’re trying to figure out the
number of possible license plate numbers in
the state. Numbers and letters are possible.
Create 7 “slots” ->
_ _ _ _ _ _ _
Ask if repeats are possible. Yes, they are.
Ask if order matters. Yes, it does. Therefore
36 36 36 36 36 36 36
= 367 = 7.836 x 1010 = 78,364,164,100
Ex. 2: What if each number and letter
was only allowed to be used once?
Using the fundamental counting principle but
saying that repeats are NOT allowed AND order
still matters:
36 35 34 33 32 31 30
= 4.207 x 1010
This set of criteria (no repeats but order
matters) is also called a permutation
Permutations
An r-permutation of a!set
of nfactorial
means
elements is an ordered Ex.
selection
of r
3! = 3∙2∙1
elements from the set of n elements
0! = 1
n!
n Pr 
n  r !
Ex. 1:How many three-letter
codes are there using letters A, B,
C, and D if no letter can be
repeated?
Note: The order does matter
4!
 24
4 P3 
1!
Combinations
The number of combinations of n
elements taken r at a time is
n!
n Cr 
r!n  r !
Order does
matter!
WhereNOT
n & r are
nonnegative integers & r < n
Ex. 3: How many committees of
three can be selected from four
people?
Use A, B, C, and D to represent the people
Note: Does the order matter?
4!
4
4 C3 
3!1!
Ex. 4: How many ways can the
4 call letters of a radio station
be arranged if the first letter
must be W or K and no letters
repeat?
2  25  24  23  27,600
Ex. 5: In how many ways can
our class elect a president, vicepresident, and secretary if no
student can hold more than one
office?
n
P3 
Ex. 6: How many five-card hands
are possible from a standard deck
of cards?
52
C5  2,598,960
Ex. 7: Given the digits 5, 3, 6, 7, 8,
and 9, how many 3-digit numbers
can be made if the first digit must
be a prime number? (can digits be
repeated?)
Think of these numbers
as if they were on tiles,
like Scrabble. After you
use a tile, you can’t use
it again.
3  5  4  60
Ex. 8: In how many ways can
9 horses place 1st, 2nd, or 3rd in
a race?
9
P3  504
Ex. 9: In how many ways can
a team of 9 players be selected
if there are 3 pitchers, 2
catchers, 6 in-fielders, and 4
out-fielders?
3
C1  2 C1  6 C4  4 C3  360