Need a total of 6 of the 12 questions answered please, it may seem long, but that is due to
the fact ive included examples to help if needed :)
In combinatorics, we work with the positive integers and the most basic of finite structures,
the finite set (e.g., {1, 2, 3} or {John, George}). Note that in a set, the order in which the
elements are listed is irrelevant and each element needs to be listed only once—that is, the
only thing that distinguishes between two sets is which objects are present and which are
not. For example, {3, 1, 2, 3} and {George, John} describe the same two sets as above.
Enumerative problems in combinatorics are usually stated as questions of the form “How
many ways are there to …?” That is, a finite set of objects or events will be described, and
we will be asked how many elements the set contains.
An ordered list is a finite sequence of objects in which order is important (i.e., there is a
first object, a second object, etc.). For example, in abcb we are concerned not only with
what objects are there, but also with how many times and in what order they occur. So bbca
and abc are ordered lists, each different from abcb and each other even though all three
lists just use the letters a, b, and c. Note that these particular examples are ordered lists of
letters although the objects in the list can theoretically be anything.
An unordered list is a finite sequence of objects in which we are concerned with how many
of each object occur, not with the order in which they are given. For example, 1, 2, 3, 1 and
1, 1, 2, 3 are considered to be the same unordered list of numbers, while 1, 2, 3 is a
different one since here the number 1 occurs only once. It might be more natural to think of
the numbers as being “types” of things, so 1, 1, 2, 3 means that we are listing two things of
type 1 and one thing of types 2 and 3. We will give some natural circumstances in which
this is a desirable type of structure below.
The observant reader may have already noticed a redundancy in the fact that a set is really
an unordered list with the restriction that we can take no more than one of each type of
available object. Similarly, an ordered list with this same restriction is usually called a
permutation. We use the special words “permutation” and “set” only because they refer to
commonly encountered objects and we want to be efficient in talking and writing about
them. Because we have this special terminology for the situation where repetition is not
allowed, we will use the general terms ordered list and unordered list to indicate the more
general situation where repetition is allowed.
4. Decide which of the four structure types (set, unordered list, permutation, or
ordered list) best characterizes the objects in each of the following situations. Can any
of the following have more than one answer?
(a)
Dealing a 13-card hand for the card games bridge or hearts
(b)
club
Selecting three officers—president, vice president, and secretary—for a
(c)
Rolling a pair of dice
Example 2
The question “How many ways can two winners be chosen for prizes from a class
consisting of just four people {Andrew, Bob, Carly, Diane}?” is the same as the
question “How many elements are in the set
{{A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D}}?
where we use only the first letter of each person’s name for brevity.
Example 6
Represent the list of all outcomes of rolling a red six-sided die and a green six-sided
die in an organized way.
SOLUTION
We use Table 5-6 with the result of the green die labeling the columns and the
result of the red die labeling the rows.
Table 5-6: Solution to Example 6
12. Answer these questions by organizing the items to be counted in any way you
like.
(a)
How many different outfits can John form if he has three shirts (red,
green, and yellow) to choose from, and two pairs of pants (black, white)?
(b)
A new board game has a standard six-sided die, and a spinner with three
colors, red, white, and blue. A player takes a turn by tossing the die and spinning
the spinner. How many different possible results are there?
Example 2
How many ways are there to order a meal consisting of one sandwich and one
beverage at a restaurant that serves five different sandwiches and six different
beverages?
SOLUTION
Call the sandwiches 1, 2, 3, 4, 5 and the beverages A, B, C, D, E, F. Then the orders
could be written as in Table 5-22. Since this is a table with six rows and five
columns, the list has a total of 30 entries.
Table 5-22:
Organizing Orders in Example 2
There is a more practical way to state the rule of products that does not involve writing
down the actual table:
Rule of products: If each entry in a list can be created by first selecting one of x objects
and then one of y objects, then the list has a total of (x)(y) entries. In terms of sets, this
means that n(A × B) = n(A) · n(B) for all finite sets A and B.
In the example above, the entries are made by first selecting one of the six objects in {A, B,
C, D, E, F} and then one of the five objects in {1, 2, 3, 4, 5}, so the list has length (5)(6) =
30
6. In how many ways can a club with 17 members elect a president, vice president,
and secretary (assuming no person can fill more than one office) for each situation
described?
(a)
There are no restrictions.
(b)
Susan has removed herself from consideration for president due to a busy
schedule, but she is willing to serve in either of the other offices.
(c)
Sam has indicated that he will serve as president only if Mary is named as
the vice president.
(d)
The club’s bylaws require that last year’s vice president becomes this
year’s president.
16. There are 16 marbles numbered 1 to 16 in a box. Marbles 1 to 5 are red, marbles 6
to 8 are green, and marbles 9 to 16 are blue. I draw out four marbles, one at a time
without replacing them, and record the result as an ordered list of four color/number
combinations. For example, I write R4, G6, R1, B10 if the marbles drawn are, in
order, #4, #6, #1, #10.
(a)
How many possible results are there?
(b)
Of these, for how many are both the first and last marbles red?
(c)
For how many are the first and second marbles different colors?
(d)
For how many are all four marbles the same color?
Example 1
How many two-element subsets of {1, 2, 3, 4} are there?
The difference between permutations and subsets is simply a matter of whether we
care about the order of the entries—in a permutation we do and in a subset we do
not. Specifically, we consider 2 3 and 3 2 to be different permutations while we
consider {2, 3} and {3, 2} to be the same set.We saw in Section 4.5 that when we
have a notion of “equivalence” on a set of objects, we call this an equivalence
relation. More important, we call a set of objects that are equivalent to one another
an equivalence class*. In this example, there is an equivalence relation on the set of
all permutations of length two with entries from {1, 2, 3, 4}. This equivalence
relation can be described as “would look the same if they were sets.” For example,
we would say that
is an equivalence class since the permutations therein would look the same if they
were sets.
Here are some other equivalence classes for this equivalence relation:
{1 3, 3 1}
{4 2, 2 4}
and others
This provides the key idea for answering the question in Example 1.
SOLUTION
(To Example 1) We know that the number of permutations of length 2 with entries
taken from {1, 2, 3, 4} is P(4, 2) = 4 · 3 = 12. If we use the equivalence relation
“would be the same if they were sets,” each equivalence class contains two
permutations, so we will have
equivalence classes. Table 5-23 shows the 12
permutations in the 6 equivalence classes. Each equivalence class (by its very
definition) identifies a single two-element subset of {1, 2, 3, 4}. In this problem, there
is a small enough number of these to simply list:
Equivalence Classes in Example 1
{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}
So the answer to the question is 6, but more important, we have developed an
approach to a more general question.
2. (a)
How many permutations of length 3 from the set {a, b, c, d, e} are there?
(b)
For the permutation acd of length 3, list all the elements of its equivalence
class for the equivalence relation “would be the same if they were sets.”
(c)
Repeat part (b) for the permutation dae.
(d)
How big is each equivalence class formed as in parts (b) and (c)?
(e)
How many subsets of {a, b, c, d, e} have three elements?
8. In how many ways can four boys and four girls stand in a circle if all the boys
stand together and all the girls stand together?
4. You are tracking your favorite baseball player by writing down his performance
in 10 successive plate appearances using an ordered list of length 10.
(a)
If you use H for a hit, S for a strikeout, B for a base-on-balls, and O for
anything else, how many results are possible?
(b)
Of the total, how many have exactly three H’s?
(c)
Of the total, how many have exactly four H’s, exactly one S, exactly two
B’s, and exactly three O’s?
Example 2
Find a recursive model for P(n, r), the number of r-permutations from {1, …, n}.
SOLUTION
The presence of two variables makes this a little more elusive, but it’s not too hard.
Imagine we already are experts on permutations of all lengths on sets of size n − 1,
and we wish to enumerate the r-permutations of {1, …, n}. We will construct these
with a two-step algorithm:
Choose an element from {1, …, n} for the first position.
Place an (r − 1)-permutation from the remaining (n − 1)-element set in the
remaining r − 1 positions.
As this algorithm will form all r-permutations from {1, …, n}, the product rule tells
us that
P(n, r) = n ⋅ P(n − 1, r − 1)
This along with the fact that P(n, 0) = 1 for all n is enough to generate any value of
P(n, r), so this is an adequate recursive model.
1. Find a recursive model for an, the number of n-digit numbers that do not use the
digit 0.
Theorem 2
For a sequence {sn} whose kth differences are constant, for all n ≥ 0,
Example 3
Use Theorem 2 to find a closed formula for the sequence
1, 5, 14, 30, 55, 91, …
SOLUTION
Table 5-32 shows the complete difference table for this sequence. The numbers in
bold are
Table 5-32:
, so according to Theorem 2,
Difference Table for Example 3
2. Find closed formulas for each of the following sequences:
(a)
s0 = 4, sn = sn−1 + 3 for all n ≥ 1
(b)
s0 = 0, sn = sn−1 + 2n for all n ≥ 1
8. A bacteria culture grows at a rate of 10% per day.
(a)
If this morning the culture has 1,000,000 bacteria, how many days will it
take for this number to double?
(b)
Suppose at the beginning of each day (excluding this morning), you
remove a sample of 50,000 bacteria for testing. Now how many days will it take for
the original 1,000,000 bacteria to double?