More Examples
2. At a local deli you can choose from six kinds of soda, 10 kinds
of sandwiches and 12 desserts. How many ways are there to
order:
a. a sandwich
b. exactly one item
c. a full meal (soda, sandwich and dessert)
d. any two items (repetition allowed)
e. any two distinct items
f. items of two different kinds
3. In Fortran, an identifier is a character string with length 1 to 7.
The first character must be a letter. How many different
identifiers are there?
Two basic variations:
1) order in which the objects are chosen or placed is significant
2) order in which the objects are chosen or placed is not significant
Permutations and Combinations
Number of orders in which the white balls can be chosen
A permutation of a set of n distinct objects is an ordered
arrangement of the n objects. If only r of the n objects are used
we have an r-permutation. Notation: P(n, r)
P(n, r) =
P(n, n) =
P(n, 0) =
Example 1: Consider the set {a, b, c, d, e, f}
# of ways to form 4-character string
# of ways to form 4-character string if no letter is repeated
Example 2: How many four-digit numbers
contain only odd digits
contain only even digits
do not contain 0
have all digits distinct
contain at least one 3
contain exactly one 3 with all digits distinct
Example—the Pick Three Game
Rules:
Ways to win:
Combinations
An r-combination is
Notation: C(n, r) or
Choose five white balls and two red balls
Combinatorial Identities
C(n, r) = C(n, n-r)
n
Σ C(n, i) = 2n
i=0
Examples
Example 1: Menu with 6 sodas, 10 sandwiches, 12 desserts. How
many ways are there to order 3 sodas, 5 sandwiches and 2
desserts, all different?
Example 2: A committee is chosen from a group of 4 men and 6
women.
a) In how many ways can a committee of size three be chosen?
b) In how many ways can a committee of size three be chosen if
there must be 1 man and 2 women?
Example 3: A student must answer 10 of 13 questions on an
exam.
How many ways can he or she take the exam if
a. there are no restrictions
b. the first two questions must be answered
c. the first or second question, but not both must be answered
d. exactly three of the first five questions must be answered
e. at least three of the first five questions must be answered
More examples
Example 4: How many five card poker hands contain
a. 2 spades and 3 red cards
b. 2 cards each of two ranks
c. all red cards with two cards each of two ranks
d. exactly one queen and exactly three spades
e. contain at least one ace?
Why won’t the following approach work?
Example 5: How many ternary strings of length 10 are there with
a. exactly two 0’s, three 1’s and five 2’s
b. exactly two 0’s, three 1’s and five 2’s with each 1 immediately
preceded by a 2.
Example 6: You invite nine friends to join you at dinner. How
many ways can the ten of you be seated around a circular table?
Inclusion/Exclusion Examples
1. Let S = {a, b, c, d, e, f}. How many strings of length 5 begin or
end with a?
2. Find the number of positive integers  100 that are even or
divisible by 7.
3. Find the number of four digit numbers that have exactly one 3.
Case 1:
Case 2:
4. How many powerball tickets have white numbers 13 or 43 but
not both?