Transcript Document

Section 9.1b…
Combinations
What are they???
With permutations, we take n objects, r at a time,
and different orderings of these objects are
considered different permutations.
In some situations, different orderings don’t matter!!!
Combinations (of n objects taken r at a time) – the
numbers of ways to select the objects, regardless of
the order in which they are arranged.
Combination Counting Formula
The number of combinations of n objects taken r
at a time is denoted n C r and is given by
n!
n Cr 
r ! n  r !
If r > n, then
for
n
0r n
Cr  0
A note about notation:
n

is commonly denoted
C
n r
 
r
Both are read:
“n choose r ”
Distinguishing Combinations
from Permutations
In each of the following scenarios, tell whether permutations
(ordered) or combinations (unordered) are being described.
Then determine the possible choices in the scenario.
1. A president, vice-president, and secretary are chosen from
a 25-member garden club.
Permutations – order matters because it matters
who gets which office.
P  13,800
25 3
Distinguishing Combinations
from Permutations
In each of the following scenarios, tell whether permutations
(ordered) or combinations (unordered) are being described.
Then determine the possible choices in the scenario.
2. A cook chooses 5 potatoes from a bag of 12 potatoes to
make a potato salad.
Combinations – the salad is the same no matter
what order the potatoes are chosen.
12 
12 C5  
  792
5
Distinguishing Combinations
from Permutations
In each of the following scenarios, tell whether permutations
(ordered) or combinations (unordered) are being described.
Then determine the possible choices in the scenario.
3. A teacher makes a seating chart for 22 students in a
classroom with 30 desks.
Permutations – a different ordering of students in
the same seats results in a different seating chart.
P

6.5787

10
30 22
27
More Guided Practice
In the Miss America pageant, 51 contestants must be narrowed
down to 10 finalists who will compete on national television. In
how many possible ways can the ten finalists be selected?
Permutations or Combinations???
51!

C
51 10 10! 51  10 !


 12, 777, 711,870
ways
More Guided Practice
The Georgia Lotto requires winners to pick 6 integers between
1 and 46. The order in which you select them does not matter;
indeed, the lottery tickets are always printed with the numbers
in ascending order. How many different lottery tickets are
possible?
Permutations or Combinations???
possible tickets

9,366,819
C
46 6
More Guided Practice
Armando’s Pizzeria offers patrons any combination of up to 10
different toppings. How many different pizzas can be ordered
(a) if we can choose any three toppings?
Order of the toppings does not matter, so:
10
C3  120
possible pizzas
More Guided Practice
Armando’s Pizzeria offers patrons any combination of up to 10
different toppings. How many different pizzas can be ordered
(b) if we can choose any number of toppings (0 through 10)?
We could add up all numbers of the form
C
10 r
for r = 0, 1,…, 10.
A quicker way: For each option (topping), we can choose either
yes or no. By the Multiplication Principle:
2  2  2  2  2  2  2  2  2  2  1024 possible
pizzas
Formula for Counting
Subsets of an n - set
n
There are 2 subsets of a set with n
objects (including the empty set and
the entire set).
More Guided Practice
A national hamburger chain used to advertise that it fixed its
hamburgers “256 ways,” since patrons could order whatever
toppings they wanted. How many toppings must have been
available?
We need to solve the following equation for n:
n
2  256
log 2  log 256
n
n log 2  log 256
log 256
n
log 2
n  8 toppings