Transcript Permutations and Combinations

In this lesson we single out two important special
cases of the Fundamental Counting Principle—
permutations and combinations.
Goal: Identity when to use permutations and
combinations.
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A permutation is one of the different
arrangements of a group of items where
order matters.
A permutation of a set of distinct objects
is an ordering of these objects.
Anytime you see “order”, plug your
numbers into the permutation equation.
Permutations give really big numbers!!
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Some permutations of the letters
ABCDWXYZ are
XAYBZWCD ZAYBCDWX
DBWAZXYC
YDXAWCZB
How many such permutations are possible?
8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320
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Its called a factorial, and it looks like an
exclamation mark (!).
 The number of permutations of n
objects is n!.
 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320
 4! = 4 x 3 x 2 x 1 = 24
How many permutations consisting of
five letters can be made from these
same eight letters? (ABCDWXYZ)
 Some are: XYZWC AZDWX AZXYB WDXZB
 By the Fundamental Counting Principle,
the number of such permutations is
8 x 7 x 6 x 5 x 4 = 6720
But there is another shortcut….
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If a set has n elements, then the number of
ways of ordering r elements from the set is
denoted by P(n, r).
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(n = the number of elements you can choose
from; r = how many you are actually going to
use)
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So, from the question of: How many
permutations consisting of five letters can be
made from these same eight letters?
(ABCDWXYZ)
P(8,5)
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A club has nine members. In how many ways
can a president, vice president, and secretary
be chosen from the members of this club?
Does order matter?
Yes, then it is a permutation, we can use the
permutation formula.
P(9, 3) =
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= 504
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From 20 raffle tickets in a hat, four tickets are
to be selected in order. The holder of the first
ticket wins a car, the second a motorcycle,
the third a bicycle, and the fourth a
skateboard. In how many different ways can
these prizes be awarded?
Does order matter?
Yes, then it is a permutation, we can use the
permutation formula.
P(20, 4) =
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P(20,4) =
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116,280
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When finding permutations, we are
interested in the number of ways of ordering
elements of a set. In many counting
problems, however, order is not important…
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A combination of r elements of a set is any
subset of r elements from the set (Order
does not matter).
If the set has n elements, then the number of
combinations of r elements is denoted by
C(n, r).
Combinations give smaller numbers!!
• The key difference between permutations and
combinations is order. If we are interested in
ordered arrangements, then we are counting
permutations; but if we are concerned with subsets
without regard to order, then we are counting
combinations.
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A coach must choose five starters from a
team of 12 players. How many different ways
can the coach choose the starters?
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How many different ways can the coach
select the 1st star, 2nd star, and 3rd star of the
game?
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A club has nine members. In how many ways
can a committee of three be chosen from the
members of this club?
Does order matter?
No, then it is a combination.
C(9,3)=
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= 84
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From 20 raffle tickets in a hat, four tickets are
to be chosen at random. The holders of the
winning tickets are to be awarded free trips
to the Bahamas. In how many ways can the
four winners be chosen?
C(20,4)=
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= 4845
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There are fourteen juniors and three seniors
in the Service Club. The club is to send four
representatives to the State Conference. How
many different ways are there to select a
group of four students to attend the
conference?
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C(17,4)=
2380
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