Transcript Chapter 13 sec 3

Chapter 13 sec. 3

Def.
 Is an ordering of distinct objects in a
straight line. If we select r different objects
from a set of n objects and arrange them in
a straight line, this is called permutation of
n objects taken r at a time.
 Order matters!!!!!
 Denoted by P(n,r)
 What
does P(5,3) mean?
 n is the number of objects from which
you may select.
 r is the number of objects that you are
selecting.

That you are counting permutations
formed by 3 different objects from
a set of five available objects.
How many permutations are there
of the letters z, r, t, and w. Write
the answer in P(n,r) notation.
 Solution:

 One way is to make a list. (too long.)
 Using the slot diagram.
Without repetition, there are 4 letters which
can be for the first position, 3 for the second,
and so on.
1st letter
2nd
3rd
4th
x
4 x 3 x 2
1

Therefore P(4,4) = 24 permutations.

Find the number of permutations. Write
it as P(n,r) notation.
 Eight objects taken three at a time.
Questions to think about.
1. How many objects (n)?
2. The number of objects being
selected (r)?
 There are 8 objects which is n.
 3 objects are being selected. (r)
 P(8,3) = 8 X 7 X 6 = 336
 n!,
called n factorial
 n•(n-1)•(n-2)•∙∙∙•2•1
 0!=1
 6!
= 6x5x4x3x2x1 = 720
 (6-3)!
 3!/4!
= 3! = 3x2x1 = 6
=(3x2x1)/(4x3x2x1)
= 1/4
 To
help you compute P(n,r)!
 P(n,r)
= n! /(n-r)!

Find the Permutation

A) 9 objects taken 4 at a time.

B) 20 objects taken 7 at a time.

C) 5 objects taken 2 at a time.



A) P(9,4) = 9!/5! = 9x8x7x6=3024
B) P(20,7) = 20!/13!= 20x19x18…x14
= 390,700,800
C) P(5,2) = 5!/3! = 20

Def.
 If we choose r objects from a set of n
objects, we say that we are forming a
combination of n objects taken r at a time.
 Notation C(n,r) = P(n,r) / r! = n! / [r!(n-r)!]
 We
are only concerned only
with choosing a set of
elements, but the order of
the elements is not
important.

This means that if the
permutations number is big, the
combination number will be smaller.

Find the Combinations

A) Eight objects taken three at a time.

B) Nine objects taken six at a time.

C) How many 3 elements sets can be
chosen from a set of 5 objects.

A) C(8,3) = 8!/(3!5!) = 8x7x6/6= 56

B) C(9,6) = 9!/(6!3!) =9x8x7/3x2=84

C) C(5,3) = 10


In the game of poker, five cards are
drawn from a standard 52-card deck.
How many different poker hands are
possible?
Solution:
 C(52,5) = 2, 598, 960

Give your answers using P(n,r) or C(n,r)
notation. The key is if order matters or not.

1. Annette has rented a summer house for
next semester. She wants to select four
roommates from a group of six friends.

2. There are 7 boats that will finish the
America’s Cup yacht race.

3. A bicycle lock has three rings with the
letters A through K on each ring. To unlock
the lock, a letter must be selected on each
ring. Duplicate letters are not allowed, and
the order in which the letters are selected on
the rings does not matter.
A)
C(6,4)
B)
P(7,7)
C)
C(11,3)