Download File - Mrs. Hille`s FunZone

Exercise
Evaluate 3!
6
Exercise
7!
Evaluate
3!
840
Exercise
7!
Evaluate
4!
210
Exercise
7!
Evaluate
3!(7 – 3)!
35
Exercise
10!
Evaluate
4!(10 – 4)!
210
Combination
A combination is a selection
of a subset of objects from a
set without regard to the
order in which they are
selected. The notation for the
number of combinations of n
distinct objects taken r at a
time is nCr .
Example 1
Identify as a permutation or
a combination: the number
of ways to place five
pictures in a line on a wall.
permutation
The order of the pictures is
important.
Example 1
Identify as a permutation or
a combination: the number
of ways to fill four eighthgrade class offices from the
seven nominees.
permutation
The office each person fills
is important.
Example 1
Identify as a permutation or a
combination: the number of
ways to choose a five-person
class party committee from
the twelve volunteers.
combination
The committee is the same
regardless of order.
Example
Identify as a permutation or
a combination: the number
of ways of choosing two
students from a group of
forty to be class
representatives.
combination
Example
Identify as a permutation or
a combination: the number
of ways of choosing two
students from a group of
forty to be the class
president and vice-president.
permutation
Formula for Combinations
To find the number of
combinations of n distinct
objects taken r at a time, use
n!
the formula nCr = r! (n – r)! .
Example 2
Use the formula to find the
number of combinations of
three books that Amos could
choose from the seven new
books that he bought.
7!
7!
=
3!4!
3!(7 – 3)!
7 × 6 × 5 × 4!
=
3 × 2 × 1 × 4!
=7×5
7C3 =
= 35 different combinations
Example 3
There are ten girls in Mrs.
Hernando’s class, and six
are to be selected for a
volleyball team. How many
different teams can be
chosen?
10!
10!
=
6!4!
6!(10 – 6)!
5
3 2
10 × 9 × 8 × 7 × 6!
=
6! × 4 × 3 × 2 × 1
=5×3×2×7
10C6 =
= 210 teams
Example
Evaluate 7C3.
35
Example
Evaluate 12C7.
792
Example
Write the answer using
combination or permutation
notation. Do not evaluate.
Example
How many different ways
can you choose five out of
seven flower types to be
included in a bouquet?
7C5
Example
How many different ways
can a leadoff hitter and a
cleanup hitter be chosen
from a group of twelve
ballplayers?
12P2
Example
Find the number of ways of
choosing two co-chairs from
a list of twelve candidates.
12C2
Example
Find the number of ways of
selecting a committee of six
men and six women from a
group of thirty men and
twenty-five women.
30C6 × 25C6
Example
How many ways can you
partition the numbers
1, 2, 3, and 4 into two sets of
two numbers each?
4C2 × 2C2
Example
How many ways are there to
divide a class of eighteen
into three equal-size reading
groups?
18C6 × 12C6 × 6C6
Example
Simplify nC1.
n
Example
Simplify nCn – 1.
n
Exercise
The school principal wants
to form a committee of five
teachers. Twelve of the
teachers in the school are
women and six are men.
Exercise
How many different
committees can be formed?
8,568
Exercise
How many different allwomen committees can be
formed?
792
Exercise
How many different all-men
committees can be formed?
6
Exercise
If there were three women on
the committee, then two men
would have to be chosen to fill
the remaining positions on the
committee. How many ways
can three women be chosen?
How many ways can two men
be chosen?
220; 15
Exercise
Using the Fundamental
Principle of Counting, find
how many ways a committee
of three women and two men
can be formed.
3,300
Exercise
How many ways can a
committee of four women
and one man be formed?
2,970