6.4 Permutations and
Combinations
Permutations:
A permutation is an ordered list of items.
Example: If we have the letters q, u, a, k, e, s,
how many 6-letter sequences are possible that
use each letter once? (Each of these
sequences is a permutation).
Factorials:
Sometimes instead of ordering all of the items
of a set, we only want a list of some of them.
A permutation of n items is an ordered list of
those items.
# of possible permutations of n items
= n factorial
= n! = n x (n - 1) x (n - 2) x … x 2 x 1
Example: How many 4-letter sequences can
be made using the letters q, u, a, k, e, s, using
each letter only once?
Note: 0! = 1
Calculate: 4!; 9!
Permutations of n items taken
r at a time.
The number of ordered lists of r items chosen
from a set of n items is given by:
P(n, r) = n x (n - 1) x (n - 2) x … x (n - r + 1)
(there are r items multiplied together)
Examples:
a. Find the number of permutations of 8 items
taken 5 at a time.
b. How many ordered sequences are possible
that contain 3 objects chosen from 7?
We can also write:
c. Calculate P(6, 2)
What if we don’t care about order?
Permutations vs. Combinations
Suppose we have 4 professors and we want to
pick 2 of them to serve on a committee. There
is no particular order! How many possible
choices do we have?
A permutation of n items taken r at a time is an
ordered list of r items chosen from n.
A combination of n items taken r at a time is an
unordered set of r items chosen from n.
Note: Lists are ordered. Sets are unordered.
Combinations of n items taken
r at a time
Example:
List all of the permutations of the letters a, b,
and c taken two at a time.
The number of combinations of n items taken r
at a time is given by:
List all of the combinations of the letters a, b,
and c taken two at a time.
or
Examples:
a. Find the number of combinations of 6 items
taken two at a time.
b. How many sets of 4 marbles can be chosen
from 6?
c. Calculate C(11,3)
More examples (finding patterns):
Calculate:
a. C(3,0) =
b. C(3,1) =
c. C(3,2) =
d. C(3,3) =
e.
f.
g.
h.
i.
C(4, 0) =
C(4, 1) =
C(4, 2) =
C(4, 3) =
C(4, 4) =
In general:
Example: Marbles
C(n, 0) =
C(n, n) =
A bag contains three red marbles, three blue
ones, three green ones, and two yellow ones
(all distinguishable from one another).
a. How many sets of 4 marbles are possible?
b. How many sets of 4 are there such that each
one is a different color?
c. How many sets of 4 are there in which at
least two are red?
C(n, 1) =
C(n, n -1) =
Example: Sequences
a. How many six-letter sequences are possible
that use the letters a, u, a, a, u, k?
b. How many six-letter sequences are possible
that use the letters f, f, a, a, f, f?
Example: More marbles
A bag contains 5 red marbles, 3 blue ones, 4
green ones, and 1 yellow one (all
distinguishable from each other).
a. How many sets of 5 marbles include all of
the blue ones?
b. How many sets of 5 marbles include at most
one of the red ones?
c. How many sets of 5 marbles include at least
one blue one but no green ones?
Example: Poker hands
Example: Poker hands
A standard deck of playing cards consists of 52
cards. Each card is in one of 13
denominations: ace (A), 2, 3, 4, 5, 6, 7, 8, 9,
10, jack (J), queen (Q), and king (K), and in one
of four suits: hearts (♥), diamonds (♦), clubs
(♣), and spades (♠).
In the card game of poker, a hand consists of 5
cards from a standard deck of 52.
a. How many different poker hands are there?
b. How many hands are there that contain two
of a kind (two of one denomination and three
of different denominations).
c. How many hands are there that contain 4 of
a kind?
d. How many hands are there that contain two
pairs?
e. How many hands contain a full house?