More on Permutations
Topics in this Section
Permutation with Repetition
Circular Permutations
Bracelet Permutations
Permutation with Repetition
n!
r1 !r2 !r3 !r4 !...rn !
How many ways are there to arrange the letters in the word
GRIFFIN?
7!
=1260
1!1! 2! 2!1!
GRIFFIN
How many different signals can be made by displaying five
flags all at a time, on a flagpole, if two are red, two are black,
and one is blue?
5!
2!2!1!
RRBBL
=30
Permutation with Repetition
n!
r1 !r2 !r3 !r4 !...rn !
How many different signals can be made by displaying five
flags all at a time, on a flagpole, if three are red, one is black,
and one is blue?
5!
=20
3!1!1!
RRRBL
In how many ways can three red, four blue, and two green
pens be distributed among nine students if each student
receives one pen?
9!
3!4!2!
=1260
RRRBBBBGG
Permutation with Repetition
n!
r1 !r2 !r3 !r4 !...rn !
How many five-digit integers can be formed using 1 and 2 twice,
and three only once?
5!
=30
2!2!1!
11223
How many ways are there to arrange the letters in the word
ALGEBRA?
7!
=2520
2!1!1!1!1!1!
AALGEBR
Circular Permutations
(n – 1)!
BUT…
How many ways are there to arrange seven chairs around a
round table?
 7  1!  6!  720
B
A
C
X
G
D
F
A
B
D
E
E
F
G
6!  720
Circular Permutations
(n – 1)!
BUT…
How many ways are there to arrange five people around a circle
if John and Bill have to sit next to each other?
A B C D J
X
B C D J
B J C D
3  2  1  2
12
Another Permutation with Restriction
Find all the permutations of the letters A, B, C, D, E, F if the
consonants must remain together.
B C D F A E
3  2  1   4  3  2  1
144
One More Permutation with Restriction
How many ways can three boys and three girls be arranged
In a circle if the girls MUST remain next to each other?
B1 B2
X B3
G1 G2 G3
3  2  1  3  2  1
36
Bracelet Permutations
n  1!  no tricks
2
How many ways are there to arrange six beads on a bracelet?
 6  1!  5!  60
2
2
How many ways are there to arrange eight charms on a
bracelet?
8  1!  7!  2520
2
2