Permutations: Arrangements of Distinct Objects
A permutation of n (distinct objects) is an arrangement of all of the objects in a definite order.
The total number of such permutations is denoted by
Note that, by the Fundamental Counting Principle,
Example 1: A choir has learned eight songs for its spring concert. In how many different ways can
the director arrange theses songs to form the concert program?
Example 2: An investment club with five members wants to select a president and a vice-president.
In how many ways can this be done?
Suppose you have n objects to choose from, but only want to select some rather than all of them.
Def.: A permutation of n (distinct) objects taken r at a time is an arrangement of r of the n objects
in a definite order. The total number of such objects is denoted by
or
Proof: Suppose we want to form an arrangement of r objects from a set of n objects.
1
2
3
4
5
r-2
r-1
Thus, by the Fundamental Counting Principal,
Ex 3: Evaluate each expression.
a) P(8,3)
b) P(6,6)
r
Ex 4: In how many ways can a chairman, treasurer, and a secretary be selected from a Board of
Directors with eight members?
Ex 5: a) Find the number of permutations of the letters of the word DIPLOMA for which the letter
L remains in the middle position.
b) How many ways are there of arranging the letters of DIPLOMA so that the letters O and I
are together?
Ex 6: At a used car lot, six cars for sale are to be parked side-by-side. In how many different ways
can this be done if:
a) the one (and only one) car with sunroof must be at the right end of the line?
b) the three black cars must be together?
Homework: Pg. 255: #4-6, 8 a,b iii, iv