Transcript 10.2 Permutations

Permutations
Objectives: Solve problems involving linear
permutations of distinct or indistinguishable
objects. Solve problems involving circular
permutations.
Standards: 2.7.8A Determine the number of
permutations for an event.
A permutation is an arrangement of objects
in a specific order.
When objects are arranged in row,
the permutation is called a linear permutation.
You can use factorial notation
to abbreviate this product:
4! = 4 x 3 x 2 x 1 = 24.
If n is a positive integer, then n factorial,
written n!, is defined as follows:
n! = n x (n-1) x (n-2) x . . . x 2 x 1.
Note that the value of 0! = 1.
I. Permutations of n Objects - the number of
permutations of n objects is given by n!
{factorial button – go to Math to PRB to # 4}
Ex 1. In 12-tone music, each of the 12 notes in an
octave must be used exactly once before any are
repeated. A set of 12 tones is called a tone row.
How many different tone rows are possible?
12! = 479,001, 600
Ex 2. How many different ways can the letters in the
word objects be arranged?
7! = 5040
II.
Permutations of n Objects Taken r at a Time –
the number of permutations of n
objects taken r at a time, denoted by P(n, r), is
given by P(n, r) = nPr =__n!_, where r < n.
(n–r)!
Ex 1. Find the number of ways to listen to 5 different CDs from
a selection of 15 CDs.
15 P 5 = 360, 360
Ex 2. Find the number of ways to listen to 4 CDs from a
selection of 8 CDs.
8 P 4 = 1680
Ex 3. Find the number of ways to listen to 3 different CDs from
a selection of 5 CDs.
5 P 3 = 60
III.
Permutations with Identical Objects –
the number of distinct permutations of
n objects with r identical objects is
given by n!/r! where 1 < r < n. The
number of distinct permutations of n
objects with r1 identical objects, r2
identical objects of another kind, r3
identical objects of another kind, . . . ,
and rk identical objects of another kind
is given by
_______n!
_.
r1 ! * r2 ! * r3 ! . . . rk !
Ex 1. Anna is planting 11 colored flowers in a line.
In how many ways can she plant 4 red flowers,
5 yellow flowers, and 2 purple flowers?
Ex 2. In how many ways can Anna plant 11 colored
flowers if 5 are white and the remaining ones are red?
11!__
(5! * 6!)
= 462
Ex 3. Frank is organizing sports equipment for the physical education room.
He has 15 balls that he must place in a line.
In how many ways can he line up 6 footballs, 2 soccer balls,
4 kickballs, and 3 basketballs?
____15!______
(6! * 2! * 4! * 3!)
= 6,306,300
Ex. 4 BETWEEN
7!
3!
= 840
III. Circular Permutations - If n distinct
objects are arranged around a circle, then
there are (n – 1)! Circular permutations of
the n objects.
Ex 2. In how many ways can seats be chosen for 12 couples
on a Ferris wheel that has 12 double seats?
(12 – 1)! = 11! = 39, 916, 800
Ex 3. In how many different ways can 17 students
attending a seminar be arranged in a circular seating pattern?
(17 – 1)! = 16! = 2.09 X 1013
Writing Activities
REVIEW OF PERMUTATIONS