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Blank sheet of paper- name
IN ORDER
Work any problem you missed with ALL
work. NO WORK = NO CREDIT
I will only grade extra sheet of paper.
Open note.
BY YOURSELF!
Permutations
and
Combinations
Permutations
A Permutation is an arrangement
of items in a particular order.
Notice,
ORDER MATTERS!
To find the number of Permutations of
n items, we can use the Fundamental
Counting Principle or factorial notation.
Permutations
The number of ways to arrange
the letters ABC:
____ ____
____
3 ____ ____
Number of choices for second blank? 3 2 ___
Number of choices for third blank?
3 2 1
Number of choices for first blank?
3*2*1 = 6
ABC
ACB
3! = 3*2*1 = 6
BAC
BCA
CAB
CBA
Permutations
To find the number of Permutations of
n items chosen r at a time, you can use
the formula
n!
n pr  ( n  r )! where 0  r  n .
5!
5!
  5 * 4 * 3  60
5 p3 
(5  3)! 2!
Permutations
Example:
A combination lock will open when the
right choice of three numbers (from 1
to 30, inclusive) is selected. How many
different lock combinations are possible
assuming no number is repeated?
Permutations
Example:
A combination lock will open when the
right choice of three numbers (from 1
to 30, inclusive) is selected. How many
different lock combinations are possible
assuming no number is repeated?
30!
30!

 30 * 29 * 28  24360
30 p3 
( 30  3)! 27!
Permutations
Example:
From a club of 24 members, a
President, Vice President, Secretary,
Treasurer and Historian are to be
elected. In how many ways can the
offices be filled?
Permutations
Example:
From a club of 24 members, a
President, Vice President, Secretary,
Treasurer and Historian are to be
elected. In how many ways can the
offices be filled?
24!
24!


24 p5 
( 24  5)! 19!
24 * 23 * 22 * 21 * 20  5,100,480
Combinations
A Combination is an arrangement
of items in which order does not
matter.
ORDER DOES NOT MATTER!
Since the order does not matter in
combinations, there are fewer combinations
than permutations. The combinations are a
"subset" of the permutations.
Combinations
To find the number of Combinations of
n items chosen r at a time, you can use
the formula
n!
C 
where 0  r  n .
n r r! ( n  r )!
Combinations
To find the number of Combinations of
n items chosen r at a time, you can use
the formula n!
C 
where 0  r  n .
n r r! ( n  r )!
5!
5!


5 C3 
3! (5  3)! 3!2!
5 * 4 * 3 * 2 * 1 5 * 4 20


 10
3 * 2 *1* 2 *1 2 *1 2
Combinations
Example:
To play dodge ball, the PE teacher
needs to pick 2 dodge balls out of a
bin that holds 7 dodge balls. How
many different ways can the PE
teacher select the dodge balls?
Combinations
To play dodge ball, the PE teacher
Example:
needs to pick 2 dodge balls out of a
bin that holds 7 dodge balls. How
many different ways can the PE
teacher select the dodge balls?
7!
7!

7C2 
2! (7  2)! 2! (5! )
Combinations
Example:
A student must answer 3 out of 5
essay questions on a test. In how
many different ways can the
student select the questions?
Combinations
Example: A student must answer 3 out of 5
essay questions on a test. In how
many different ways can the
student select the questions?
5!
5! 5 * 4


 10
5 C3 
3! (5  3)! 3!2! 2 * 1
Combinations
Example:
A basketball team consists of two
centers, five forwards, and four
guards. In how many ways can the
coach select a starting line up of
one center, two forwards, and two
guards?
Combinations
Example:
Center:
A basketball team consists of two centers, five forwards,
and four guards. In how many ways can the coach select a
starting line up of one center, two forwards, and two
guards?
Forwards:
Guards:
2!
5! 5 * 4
4! 4 * 3
C



10
C


2

6
5 2
2
1
4 C2 
2!3! 2 * 1
2!2! 2 * 1
1!1!
2
C1 * 5 C 2 * 4 C 2
Thus, the number of ways to select the
starting line up is 2*10*6 = 120.
Homework
Worksheet