Transcript 12.4 – Permutations & Combinations

12.4 – Permutations &
Combinations
• Permutation – all possible arrangements of
objects in which the order of the objects is
taken in to consideration.
• Permutation – all possible arrangements of
objects in which the order of the objects is
taken in to consideration.
Ex. 1 A travel agency is planning a vacation
package in which travelers will visit 5 cities
around Europe. How many ways can the agency
arrange the 5 cities along the tour?
• Permutation – all possible arrangements of
objects in which the order of the objects is
taken in to consideration.
Ex. 1 A travel agency is planning a vacation
package in which travelers will visit 5 cities
around Europe. How many ways can the agency
arrange the 5 cities along the tour?
5 ∙ (5-1) ∙ (5-2) ∙ (5-3) ∙ (5-4)
• Permutation – all possible arrangements of
objects in which the order of the objects is
taken in to consideration.
Ex. 1 A travel agency is planning a vacation
package in which travelers will visit 5 cities
around Europe. How many ways can the agency
arrange the 5 cities along the tour?
5 ∙ (5-1) ∙ (5-2) ∙ (5-3) ∙ (5-4)
5∙ 4 ∙ 3 ∙ 2 ∙ 1
• Permutation – all possible arrangements of
objects in which the order of the objects is
taken in to consideration.
Ex. 1 A travel agency is planning a vacation
package in which travelers will visit 5 cities
around Europe. How many ways can the agency
arrange the 5 cities along the tour?
5 ∙ (5-1) ∙ (5-2) ∙ (5-3) ∙ (5-4)
5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120
• Permutation – all possible arrangements of
objects in which the order of the objects is
taken in to consideration.
Ex. 1 A travel agency is planning a vacation
package in which travelers will visit 5 cities
around Europe. How many ways can the agency
arrange the 5 cities along the tour?
5 ∙ (5-1) ∙ (5-2) ∙ (5-3) ∙ (5-4)
5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120
*This is called factorial, represented by “!”.
• Permutation – all possible arrangements of
objects in which the order of the objects is
taken in to consideration.
Ex. 1 A travel agency is planning a vacation
package in which travelers will visit 5 cities
around Europe. How many ways can the agency
arrange the 5 cities along the tour?
5 ∙ (5-1) ∙ (5-2) ∙ (5-3) ∙ (5-4)
5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120
*This is called factorial, represented by “!”.
5! = 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120
Permutation Formula – The number of
permutations of n objects taken r at a time is the
quotient of n! and (n – r)!
Permutation Formula – The number of
permutations of n objects taken r at a time is the
quotient of n! and (n – r)!
P(n,r) =
n!
(n – r)!
Permutation Formula – The number of
permutations of n objects taken r at a time is the
quotient of n! and (n – r)!
P(n,r) =
n!
(n – r)!
Ex. 2 The librarian is placing 6 of 10 magazines
on a shelf in a showcase. How many ways can
she arrange the magazines in the case?
Permutation Formula – The number of
permutations of n objects taken r at a time is the
quotient of n! and (n – r)!
P(n,r) =
n!
(n – r)!
Ex. 2 The librarian is placing 6 of 10 magazines
on a shelf in a showcase. How many ways can
she arrange the magazines in the case?
P(n,r) =
n!
(n – r)!
Permutation Formula – The number of
permutations of n objects taken r at a time is the
quotient of n! and (n – r)!
P(n,r) =
n!
(n – r)!
Ex. 2 The librarian is placing 6 of 10 magazines on a
shelf in a showcase. How many ways can she
arrange the magazines in the case?
P(n,r) =
n!
(n – r)!
P(10,6) =
10!
(10 – 6)!
Permutation Formula – The number of permutations of n
objects taken r at a time is the quotient of n! and (n – r)!
P(n,r) =
n!
(n – r)!
Ex. 2 The librarian is placing 6 of 10 magazines on a shelf
in a showcase. How many ways can she arrange the
magazines in the case?
P(n,r) =
n!
(n – r)!
P(10,6) =
10!
(10 – 6)!
P(10,6) = 10!
4!
Permutation Formula – The number of permutations of n
objects taken r at a time is the quotient of n! and (n – r)!
P(n,r) =
n!
(n – r)!
Ex. 2 The librarian is placing 6 of 10 magazines on a shelf in a
showcase. How many ways can she arrange the magazines in
the case?
P(n,r) =
n!
(n – r)!
P(10,6) = 10!
(10 – 6)!
P(10,6) = 10!
4!
P(10,6) = 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1
4∙3∙2∙1
Permutation Formula – The number of permutations of n
objects taken r at a time is the quotient of n! and (n – r)!
P(n,r) =
n!
(n – r)!
Ex. 2 The librarian is placing 6 of 10 magazines on a shelf in a
showcase. How many ways can she arrange the magazines in
the case?
P(n,r) =
n!
(n – r)!
P(10,6) = 10!
(10 – 6)!
P(10,6) = 10!
4!
P(10,6) = 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1
4∙3∙2∙1
Permutation Formula – The number of permutations of n
objects taken r at a time is the quotient of n! and (n – r)!
P(n,r) =
n!
(n – r)!
Ex. 2 The librarian is placing 6 of 10 magazines on a shelf in a
showcase. How many ways can she arrange the magazines in
the case?
P(n,r) =
n!
(n – r)!
P(10,6) = 10!
(10 – 6)!
P(10,6) = 10!
4!
P(10,6) = 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1
4∙3∙2∙1
P(10,6) = 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 = 151,200
• Combinations – a selection of objects in which
order is not considered.
• Combinations – a selection of objects in which
order is not considered.
Combination Formula – The number of
combinations of n objects taken r at a time is the
quotient of n! and (n – r)!r!
• Combinations – a selection of objects in which
order is not considered.
Combination Formula – The number of
combinations of n objects taken r at a time is the
quotient of n! and (n – r)!r!
C(n,r) =
n!
(n – r)!r!
Ex. 3 Horatio works part-time at a local
department store. His manager asked him to
choose for display 5 different styles of shirts
from the wall of the store that has 8 shirts on it
to put in a display. How many ways can he
choose the shirts?
Ex. 3 Horatio works part-time at a local
department store. His manager asked him to
choose for display 5 different styles of shirts
from the wall of the store that has 8 shirts on it
to put in a display. How many ways can he
choose the shirts?
C(n,r) =
n!
(n – r)!r!
Ex. 3 Horatio works part-time at a local
department store. His manager asked him to
choose for display 5 different styles of shirts
from the wall of the store that has 8 shirts on it
to put in a display. How many ways can he
choose the shirts?
C(n,r) =
n!
(n – r)!r!
C(8,5) = 8!
(8 – 5)!5!
Ex. 3 Horatio works part-time at a local
department store. His manager asked him to
choose for display 5 different styles of shirts
from the wall of the store that has 8 shirts on it
to put in a display. How many ways can he
choose the shirts?
C(n,r) =
n!
(n – r)!r!
C(8,5) = 8!
(8 – 5)!5!
C(8,5) = 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1
3∙2∙1∙5∙4∙3∙2∙1
Ex. 3 Horatio works part-time at a local
department store. His manager asked him to
choose for display 5 different styles of shirts
from the wall of the store that has 8 shirts on it
to put in a display. How many ways can he
choose the shirts?
C(n,r) =
n!
(n – r)!r!
C(8,5) = 8!
(8 – 5)!5!
C(8,5) = 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 56
3∙2∙1∙5∙4∙3∙2∙1