Transcript 16.3 Distinguishable Permutations

3. Suppose you take 4 different
routes to Trenton, the 3 different
routes to Philadelphia.
How many different ________ • _________
Trenton Philadelphia
routes can you take
___4____ • ___3_____
for the trip to
Philadelphia by way
12
of Trenton?
4. You have 10 pairs of pants, 6
shirts, and 3 jackets.
How many outfits
can you have
consisting of a
shirt, a pair of
pants, and a
jacket?
______•______•______
Shirts Pants Jackets
___6__•__10__•__3___
180
5. Fifteen people line up for
concert tickets.
a) How many
different
arrangements are
possible?
15•14•13•12•11•10•9•8•
7•6•5•4•3•2•1 =
1,307,674,368,000
b) Suppose that a
certain person must
be first and another
person must be last.
How many
arrangements are now
possible?
1•13•12•11•10•9•8•
7•6•5•4•3•2•1•1 =
6,227,020,800
6) Using the letters A, B, C, D, E, F
a) How many “words”
can be made using all 6
letters?
6 • 5 • 4 • 3 • 2 • 1 = 720
b) How many of these
words begin with E ?
1 • 5 • 4 • 3 • 2 • 1 = 120
c) How many of these
words do NOT begin
with E? 720 –120 = 600
d) How many 4-letter
words can be made if
no repetition is allowed?
6•5•4•3 = 360
e) How many 3-letter
words can be made if
repetition is allowed?
6 • 6 • 6 = 216
f) How many 2 OR 3
letter words can be
made if repetition is
not allowed?
6•5+6•5•4 = 30 + 120 = 150
g) If no repetition is
allowed, how many
words containing at
least 5 letters can be
made? (both letter 6a)
720 + 720 = 1440
16.3 Distinguishable
Permutations
OBJ:  To find the quotient of numbers
given in factorial notation
 To find the number of
distinguishable permutations
when some of the objects in an
arrangement are alike
EX:  Find the value of 8! _
4! x 3!
One Method
Short Method
8•7•6•5•4•3•2•1
4•3•2•1•3•2•1
8 • 7 • 6 • 5 • 4!
4! • 3 • 2 • 1
1680
6
280
EX:  Find the value of
Short Method
6 • 5 • 4!
4! • 2 • 1
30
2
15
6! _
4! x 2!
EX:  Find the value of
Short Method
12 • 11 • 10 • 9!
3 • 2 • 1 • 9!
1320
6
220
12! _
3! x 9!
NOTE: The letters in the word Pop are
distinguishable since one of the two p’s is a
capital letter. There are 3!, or 6,
distinguishable permutations of P, o, p.
Pop Ppo oPp opP poP pPo
In the word pop, the two p’s are alike and can
be permuted in 2! ways. The number of
distinguishable permutations of p, o, p is
3! , or 3.
2!
pop
ppo
opp
The number of distinguishable permutations of
the 5 letters in daddy is 5!
3! since the three d’s
are alike and can be permuted in 3! ways.
DEF:  Number of
Distinguishable Permutations
Given n objects in which a of them are
alike, the number of distinguishable
permutations of the n objects is n!
a!
EX:  How many distinguishable
permutations can be formed from
the six letters in pepper?
6!__
3! • 2!
6 • 5 • 4 • 3!
3! • 2 • 1
60
EX:  How many distinguishable
six- digit numbers can be formed
from the digits of 747457?
6!__
3! • 2!
6 • 5 • 4 • 3!
3! • 2 • 1
60
EX:  How many distinguishable
signals can be formed by displaying
eleven flags if 3 of the flags are red, 5
are green, 2 are yellow, and 1 is white?
11!______
3! • 5! • 2! • 1!
11 • 10 • 9 • 8 • 7 •6 •5!
3 • 2 • 1 •5! • 2 •1 •1
332640
12
27720
16.4 Circular Permutations
OBJ:  To find the number of
possible permutations of
objects in a circle
NOTE: Three objects may be arranged in
a line in 3!, or 6, ways. Any one of the
objects may be placed in the first position
ABC ACB BAC BCA CAB CBA
In a circular permutation of objects,
there is no first position. Only the
positions of the objects relative to
one another are considered.
EX:  In the figures below, Al, Betty and
Carl are seated in a circular position with
each person facing the center of the
circle.
In each of the first three figures, Al has Betty
to his left and Carl to his right. This is one
circular permutation of Al, Betty, and Carl.
A
C
B
C
B
A
B
A
C
The remaining three figures each show Al with
Betty to his right and Carl to his left. Again, these
count as only one circular permutation of the three
A
B
C
B
C
A
C
A
B
DEF: Number of Circular
Permutations
The number of circular permutations of n
distinct objects is (n-1)!
EX:  A married couple invites 3 other
couples to an anniversary dinner. In how
many different ways can all of the 8
people be seated around a circular table?
(8 – 1)!
7!
5040
7. How many distinguishable
permutations can be made using
all the letters of:
a) GREAT
5•4•3•2•1
5! = 120
b) FOOD
4! = 4 • 3 • 2!
2!
2!
12
c) TENNESSEE
9!
4! 2! 2!1!
9 • 8 • 7 • 6 • 5 • 4!
4! 2 • 2
15,120
4
= 3,780