Transcript Combinations.

Combinations
Combinations
Objectives:
(1) Students will be able
combinations to find all
arrangements
involving
a
number of choices.
to use
possible
limited
Essential Questions:
(1) What are combinations and how can
we find them?
Combinations
What is a Combination?
- Have you ever played a sport and thought
about how many different ways the
coach could have assigned players to the
starting lineup?
- A COMBINATION is an arrangement or
listing in which order IS NOT important.
Combinations
How Do I Find The Value of A Combination?
- We calculate the value of a combination
in the following way:
C(5,3) =
Combinations
How Do I Find The Value of A Combination?
- We calculate the value of a combination
in the following way:
Start with this number
5
x
4
x
3
60
C(5,3) =
=
= 10
3x2x1
6
Count down this many numbers
Combinations
Real World Example:
If Coach Bob McKillop has 12 basketball
players on his team, how many ways can
he choose 5 players to start a game?
Combinations
Real World Example:
If Coach Bob McKillop has 12 basketball
players on his team, how many ways can
he choose 5 players to start a game?
P(12,5) 12 x 11 x 10 x 9 x 8
C(12,5) =
=
5!
5x4x3x2x1
95040
=
= 792 ways
120
Combinations
Example 1: Combinations.
Find the value of C(6,3).
Combinations
Example 1: Combinations.
Find the value of C(6,3).
C(6,3) =
P(6,3)
3!
=
6x5x4
3x2x1
=
120
6
= 20
Combinations
Example 2: Combinations.
Find the value of C(15,2).
Combinations
Example 2: Combinations.
Find the value of C(15,2).
C(15,2) =
P(15,2)
2!
=
15 x 14
2x1
=
210
2
= 105
Combinations
Example 3: Election Candidates.
How many ways can a delegation of 4
people be selected from a class of 22
students?
Combinations
Example 3: Election Candidates.
How many ways can a delegation of 4
people be selected from a class of 22
students?
C(22,4) =
P(22,4)
4!
=
=
22 x 21 x 20 x 19
175,560
24
4x3x2x1
=
= 7315 ways
Combinations
Example 4: Birthday Party.
Sommer is having a birthday party. She
has narrowed the list to 9 people, but
she can only take 4.
How many
combinations of friends are possible?
Combinations
Example 4: Birthday Party.
Sommer is having a birthday party. She
has narrowed the list to 9 people, but
she can only take 4.
How many
combinations of friends are possible?
C(9,4) =
P(9,4)
4!
=
9x8x7x6
4x3x2x1
=
3024
24
= 126 ways
Combinations
Real World Example: Taste Test.
A taste test of 9 different soft drinks is
held at Ferndale. If each taster is
randomly given 5 of the drinks to taste,
how many combinations of soft drinks
are possible?
Combinations
Real World Example: Taste Test.
A taste test of 9 different soft drinks is
held at Ferndale. If each taster is
randomly given 5 of the drinks to taste,
how many combinations of soft drinks
are possible?
C(9,5) =
P(9,5)
5!
=
9x8x7x6 x5
5x4x3x2x1
=
15120
120
= 126 ways
Combinations
Guided Practice: Find the value.
(1) C(8,3) = ?
(2) How many three card hands can be
dealt from a deck of 52 cards?
Combinations
Guided Practice: Find the value.
(1) C(8,3) = 56
(2) How many three card hands can be
dealt from a deck of 52 cards?
22,100 different ways
Combinations
Independent Practice: Find the value.
(1) C(6,4) = ?
(2) How many ways can you choose four
items from a Chinese menu of 14 items?
Combinations
Independent Practice: Find the value.
(1) C(6,4) = 15
(2) How many ways can you choose four
items from a Chinese menu of 14 items?
1001 different ways
Combinations
Real World Example: Quiz Questions.
On a English quiz you are allowed to answer
4 out of the 6 six questions. How many
ways can you choose the questions?
Combinations
Real World Example: Quiz Questions.
On a English quiz you are allowed to answer
4 out of the 6 six questions. How many
ways can you choose the questions?
C(6,4) =
P(6,4)
4!
=
6x5x4x3
4x3x2x1
=
360
24
= 15 ways
Combinations
Summary:
- Combinations involve arrangements or
listings where order is not important.
- We use the following notation:
C(9,4) =
* The symbol C(9,4) represents the number of combinations of
9 possible things to take, and we are taking 4 of them
Combinations
Summary:
- Combinations involve arrangements or
listings where order is not important.
- We use the following notation:
Start with this number
C(9,4) =
Combination
9x8x7x6
4!
=
9x8x7x6
4x3x2x1
Count down this many numbers
= 126
Combinations
Homework: