Intensive Algebra 2
Name________________________
Counting & Order: Permutations vs. Combinations
April 13, 2011
Permutations:
Suppose we are given a set of distinct objects. When we order these objects in such a
manner such that order matters, we call this ordering a permutation of the set.
As an example, suppose you are given a set of letters, ABCDEF. Below are some
permutations of this set:
ABCDEF
DEAFBC
FACEDB
CABFED
While the elements of the set remain the same, the order of these elements is distinct
and therefore each of the above is a unique permutation.
Q: Given the set of letters above (ABCDEF), how many permutations exist using all 6 letters?
Q: Suppose we work with the same set of letters, but instead we ask how many 5-letter
permutations exist?
Q: Suppose we work with the same set of letters, but instead we ask how many 4-letter
permutations exist?
Using the answers above as a guide, when we want to address the number of
permutations of n objects taken r at a time, we write:
P ( n, r ) 
Consider a rather interesting example:
Eight workers are cleaning a large house. Five are needed to clean windows, two to
clean the carpets, and one to clean the rest of the house. In how many different
ways can these tasks be assigned to the eight workers?
Hint: If a set of n objects consists of k different kinds of objects with n1 objects of
the first kind, n2 objects of the second kind, n3 objects of the third kind, and so on,
where n1 + n2 +…+ nk = n, the number of distinguishable permutations of these objects
is:
n!
n1 ! n2 ! n3 ! nk !
Now, apply this to the last example and see that this is a more efficient manner for
arriving at the number of distinguishable permutations.
Combinations:
Suppose we are given a set of distinct objects. When we order these objects in such a
manner such that order does not matter, we call this ordering a combination of the set.
As an example, suppose you are given a set of letters, ABCD.
Q: Given the set of letters above, how many combinations exist using all 4 letters?
Q: Suppose we work with the same set of letters, but instead we ask how many 3-letter
combinations exist?
Q: Suppose we work with the same set of letters, but instead we ask how many 2-letter
combinations exist?
Using the answers above as a guide, when we want to address the number of
combinations of n objects taken r at a time, we write:
C ( n, r ) 
So, order DOES NOT MATTER for combinations and order MATTERS for permutations…
Hence, it should make sense then that the number of permutations will at least be as
large (almost always larger) than the number of combinations. The manner in which we
calculate the number of each should also reflect this idea.
Exercises
Directions: for each question, first explain why you chose either a permutation or a
combination to count, then give your answer.
1) In how many different ways can a president, vice president, and secretary be chosen
from a class of 15 students?
2) A ship carries five signal flags of different colors. How many different signals can be
sent by hoisting exactly three of the five flags on the ship’s flagpole in different orders?
3) In how many different ways can two blue marbles and four red marbles be arranged
in a row?
4) A jogger jogs every morning to his health club, which is eight blocks east and five
blocks north of his home. He always takes a route that is as short as possible, but he
likes to vary it (he lives in a neighborhood where the streets were constructed on a
“grid” model). How many different routes can he take?
5) How many different five-card hands can be dealt from a deck of 52 cards?
6) In the California Lotto game, a player chooses six numbers from 1 to 53. It costs $1 to
play this game. How much would it cost to buy every possible combination of six
numbers to ensure picking the winning numbers?
7) A class has 20 students, of which 12 are female and 8 are males. In how many ways
can a committee of five students be picked form this class under each condition?
a) No restriction is placed on the number of males or females of the committee.
b) No males are to be included on the committee.
c) The committee must have three females and 2 males.