Transcript Section 3-6 - Gordon State College

Section 3-6
Counting
FUNDAMENTAL COUNTING
RULE
For a sequence of two events in which the first
event can occur m ways and the second event can
occur n ways, the events together can occur a
total of m · n ways.
This generalizes to more than two events.
EXAMPLES
1. How many two letter “words” can be formed if the
first letter is one of the vowels a, e, i, o, u and the
second letter is a consonant?
2. OVER FIFTY TYPES OF PIZZA! says the sign as
you drive up. Inside you discover only the choices
“onions, peppers, mushrooms, sausage, anchovies,
and meatballs. There are also 3 different types of
crust and 4 types of cheese. Did the advertisement
lie?
3. Janet has five different books that she wishes to
arrange on her desk. How many different
arrangements are possible?
4. Suppose Janet only wants to arrange three of her
five books on her desk. How many ways can she
FACTORIALS
n! n  (n  1)  (n  2)    3  2 1
NOTE: 0! is defined to be 1. That is, 0! = 1
FACTORIAL RULE
A collection of n objects can be arranged in order
n! different ways.
PERMUTATIONS
A permutation is an ordered arrangement.
A permutation is sometimes called a sequence.
PERMUTATION RULE
(WHEN ITEMS ARE ALL
DIFFERENT)
The number of permutations (or sequences) of r
items selected from n available items (without
replacement) is denoted by nPr and is given by the
formula
n!
n Pr 
(n  r )!
PERMUATION RULE
CONDITIONS
• We must have a total of n different items
available. (This rule does not apply if some
items are identical to others.)
• We must select r of the n items without
replacement.
• We must consider rearrangements of the same
items to be different sequences. (The
arrangement ABC is the different from the
arrangement CBA.)
EXAMPLE
Suppose 8 people enter an event in a swim meet.
Assuming there are no ties, how many ways
could the gold, silver, and bronze prizes be
awarded?
PERMUTATION RULE
(WHEN SOME ITEMS ARE
IDENTICAL TO OTHERS)
If there are n items with n1 alike, n2 alike, . . . ,
nk alike, the number of permutations of all n
items is
n!
(n1!)(n2!) (nk !)
EXAMPLE
How many different ways can you rearrange the
letters of the word “level”?
COMBINATIONS
A combination is a selection of objects without
regard to order.
COMBINATIONS RULE
The number of combinations of r items selected
from n different items is denoted by nCr and is
given by the formula
n!
n Cr 
(n  r )! r!
NOTE: Sometimes nCr is denoted by
n
  .
r
COMBINATIONS RULE
CONDITIONS
• We must have a total of n different items
available.
• We must select r of those items without
replacement.
• We must consider rearrangements of the same
items to be the same. (The combination ABC
is the same as the combination CBA.)
EXAMPLES
1. From a group of 30 employees, 3 are to be
selected to be on a special committee. In
how many different ways can the employees
be selected?
2. If you play the New York regional lottery
where six winning numbers are drawn from
1, 2, 3, . . . , 31, what is the probability that
you are a winner?
3. Exercise 34 on page 183, [15-18] page 188.
EXAMPLE
Suppose you are dealt two cards from a wellshuffled deck. What is the probability of being
dealt an “ace” and a “heart”?
PERMUTATIONS VERSUS
COMBINATIONS
When different orderings of the same items are
to be counted separately, we have a
permutation problem, but when different
orderings are not to be counted separately, we
have a combination problem.