42 Factorials and Permutations

1
4-2 Factorials and Permutations
2
Imagine 3 animals running a race

• How many different finish orders could there be?

FINISH
D
H
S
3
1st
2nd
3rd
Order
S
DHS

• D

H
DSH
S
HDS
H
HSD
D
D
SHD
S
H
SDH
4
THEREFORE

• There are 6 possible permutations (ordered lists)
  for the race.
• This technique will be too cumbersome for
  questions with any complexity
• So.

5
Another way

• 1st

2nd
3rd
How many choices are there for first place?
3 X 2 X 1 6 (This can be compressed even
further)
Second place?
Third place?
6
Note

• 3 X 2 X 1 can be compressed into Factorial
  Notation 3!

n! n X (n 1) X (n 2) X X 3 X 2 X
1
Ex 5! 5 X 4 X 3 X 2 X 1 120
7
Simplify (on board)

• 8!

8
The senior choir has a concert coming up where
they will perform 5 songs. In how many different
orders can they sing the songs?
9
In how many ways could 10 questions on a test be
arranged if a) there are no limitationsb) the
Easiest question and the most Difficult question
are side by sidec) E and D are never side by
side
10

• a) No limitations

9
8
7
6
5
3
2
1
10!
10
4
b) E and D are side by side
D

9!
E
X 2
c) E and D are never side by side
10! 9! X 2
11
Permutation (when order matters)

• A permutation is an ordered arrangement of
  objects (r) selected from a set (n).

12

• P(n,r) (also written as nPr) represents the
  number of permutations possible in which r
  objects from a set of n different objects are
  arranged.
• With the 3 animal race, it would have been 3
  objects (n 3), permute 3 objects (r 3)
• P(3,3) or 3P3

13
How many first, second, and third place finishers
can there be with 5 animals?

• 5

4
3
1st
2nd
3rd
5 X 4 X 3 60 (way too many to tree) P(5,3) or
5P3
14
We want to use the factorial notation.5
animals, 3 spots
1
1

• 5 X 4 X 3 X 2 X 1

5!
2!
2 X 1
1
1
5!
5 X 4 X 3
(5 3)!
60
n!
P(n,r)
(n r)!
15
How many different sequences of 13 cards can be
drawn from a deck of 52?

• 52P13

52P13
16

• Pg 239
• 1-4 odd
• 7,9,10,11
• 14,15,19,20