6-4 Permutations and Combinations
FACTORIAL
n! = n  (n-1)  (n – 2)  …  2  1
e.g., 4! = 4  3  2  1
Note: 4! = 4  3! In general, n! = n  (n – 1)!
Permutation:
 a selection of a pre-specified number of items from a given
set of items, where
 different orderings of the same items are considered to be
different selections (and are counted separately)
Example:
set of items = {red, orange, yellow, green}
some permutations, selecting 3 at a time:
(red, orange, yellow)
(orange, red, yellow)
(yellow, green, red)
(green, yellow, orange)
 are there any more? how many are there in all?
 you don’t need to list them all and then count them
Here’s a formula (based on the multiplication principle):
The number of permutations of n things taken r at a time is
denoted by nPr, or Pn,r and the formula is:
nPr
= n  (n - 1)  (n-2)…
(for r factors)
(formula different from book’s but equivalent and easier to use)
To answer the above question:
The number of permutations of 4 things taken 3 at a time:
4P3 = 4×3×2 = 24
6-4
p. 1
Combination:
 a selection of a pre-specified number of items from a given set
of items, where
 different orderings of the same items are considered to be the
same combination (and are not counted separately)
Example:
set of items = {r, o, y, g}
some combinations, selecting three at a time:
(y, r, o) (r, o, g) (r, o, y) NOPE!!! already listed!
 are there any more? how many are there?
 think about (r, y, g), (y, r, g), etc.
 all counted separately as permutations
 now counting once as combinations
 how many (r, y, g), (y, r, g) etc. pairs are there?
 use permutation formula: 3 P3 = 321 = 6
 so permutations count 6 times as many outcomes as
combinations do
 hence, we need to divide by 6 to get #combinations
4  3 2
 answer:
=4
3  2 1
The number of combinations of n things taken r at a time is
denoted by nCr, or Cn,r and the formula is:
r factors


n  (n  1)  (n  2)...
nCr =
r  (r  1)  (r  2)...


r factors
(formula different from book’s but equivalent and easier to use)
6-4
p. 2