Combinations
A combination is a grouping of
things
ORDER DOES NOT MATTER
How many arrangements of the letters a, b, c
and d can we make using 3 letters at a time if
order does not matter? We know there are 4!
= 24 permutations. Listed out they are:
abc
abd
acd
bcd
acb
adb
adc
bdc
bac
bad
cad
cbd
bca
bda
cda
cdb
cab
dab
dac
dbc
cba = 1 combination
dba = 1 combination
dca = 1 combination
dcb = 1 combination
= 4 combinations, total
Each of the above combinations had 3
letters, so there were 3! ways to change
the order around (3! permutations). If
order doesn’t matter we will divide by that
number.
The number of ways of choosing r objects from a set
of n without regard to order is:
Pr
n!

n Cr 
r! r!(n  r )!
n
This is commonly read as “n choose r”
Two examples to show the difference
between permutations and combinations:
How many seating arrangements of 6
students can be made from a class of
30? (order matters – a permutation)
30!
 427,518,000
30 P6 
(30  6)!
How many ways are there of choosing 6
students for a class project in a class of 30?
(order does not matter – just that 6
students are picked – a combination)
30!
C


593
,
775
30 6
6!(30  6)!
How many different 6 number lottery
tickets can be issued? A purchaser
picks 6 numbers from 00 – 99 and it
does not matter which order they are
in.
Picking 6 correct lottery numbers . . .
100 numbers to pick from
Want the 6 that are correct
100!
 1,192,052,400
100 C6 
6!(100  6)!
How many different 5-card
poker hands are there?
Different 5-card poker hands . . .
52 cards to pick from
Want 5 cards total
52!
 2,598,960
52 C5 
5!47!
How many different 5-card
hands can there be if all cards
must be clubs (a flush in clubs)?
A flush in clubs . . .
13 clubs to pick from
Want 5 cards total
13!
 1287
13 C5 
5!(13  5)!
How many different 5-card
hands can there be if all cards
must be the same suit (a flush
in any suit)?
A flush in ANY suit . . .
4 cases that are the same - the 4
cases are from the 4 suits: hearts,
spades, clubs or diamonds.
Same as the previous example

13! 
  5148
413 C5   4
 5!(13  5)! 
How many different 5-card hands
can there be that contain exactly 3
Aces?
5 cards with exactly 3 aces . . .
# ways to
get 3 Aces
4
*
C3 *
# ways to
get other 2
cards
48
C2
4!
48!

*
 4512
3!1!
2!46!
How many different 5-card
hands can there be that contain
at least 3 Aces?
5 cards with at least 3 aces . . .
# ways to get
3 Aces (from
previous
example)
# ways to get
4 Aces
+
4
4512 
C4 *
48
C1
4!
48!
*
4!0!
1!47!
= 4512 + 48 = 4560 ways
How many different 5-card
hands can there be that contain
exactly 2 Hearts?
5 cards with exactly 2 hearts . . .
# ways to
get 2 Hearts
13
C2
*
*
# ways to get
other 3 cards
39
C3
13!
39!

*
 712,842
2!11!
3!36!