Name
Due: Monday, February 13
Homework 7
1. The following question refers to the maze below.
(a) In how many ways can you travel from A to C in this maze if you must always travel from
left to right, and you cannot travel backwards?
(b) There is a road block along the maze so that you cannot pass the red dot. In how many ways
can you now travel from A to C in this maze if you must always travel from left to right, and
you cannot travel backwards?
2. (a) Explain a similarity or difference between a permutation and combination. State clearly if
you are discussing a similarity or difference.
(b) State the (Generalized) Multiplication Principle.
(c) State the inclusion/exclusion principle.
(d) What are DeMorgan’s Laws? (Hint: There are two.)
(e) Use a Venn Diagram to show one of DeMorgan’s Laws.
3. (a) Given that the set S has 10 elements, the set T has 10 elements, and S ∩ T has 5 elements,
how many elements does S ∪ T have?
(b) Given that S 0 = {1, α, β, 4} and T 0 = {1, A, 5}, list the elements of (S ∩ T )0
(c) Given the following Venn diagram with the sizes of its basic regions filled in, how many
elements does B ∪ C have?
4. The following table gives some data about students at Notre Dame. An experiment consists of
choosing one of the people from this list.
Name
Hometown
Bob
Kansas City
Mary Minneapolis
Jane
Orlando
Orlando
Aaron
Henry Minneapolis
Anna Kansas City
Amber Minneapolis
Year
Freshman
Freshman
Senior
Junior
Junior
Sophomore
Junior
(a) Let A = {Juniors}. List the elements in A.
(b) Let B = {People from Minneapolis}. List the elements in B.
(c) List the elements in A ∩ B. Give a verbal description of A ∩ B. (DO NOT just say ”A
intersect B”)
(d) Let C = {People from Kansas City}. List the elements in (A ∩ B) ∪ C
(e) What is A ∩ (B ∩ C)?
5. (a) Shade the indicated subset
i. A
ii. A ∪ A0
iii. A ∩ A0
iv. (A ∪ B)0
v. A ∩ C
vi. A ∪ (B ∪ C)0
6. Write the following as a SINGLE multinomial or binomial coefficient.
(a) What is the coefficient of x3 y 3 in the expansion of (x + y)6 ?
(b) The number of ways to choose 3 pieces of candy out of a bowl of 20 pieces of candy.
(c) The number of 7 letter words (strings of letters) which consist of one B, three A’s, two N’s
and one S.
(d) The number of unsorted bridge hands. (A bridge hand consists of 13 cards from a standard
deck of 52).
(e) In summer camp 20 children are split into groups for activities. 10 play kickball, 3 go
canoeing, 3 go hiking, and 4 play tennis. How many ways can these groups be made?
7. For Reliable Airlines, flight codes are determined by one digit followed by two letters followed by
four digits.
(a) How many such flight codes are possible?
(b) If Reliable were to exclude all zeros for the digits, how many flight codes would be possible?
(c) Ignore part 7b for this problem. Suppose Reliable Airlines were to include a two letter code
of the flights landing nation before the first digit of the code and Reliable Airlines lands in
15 countries. How many codes are possible? For example France’s code is ’FR’, so a possible
code for a flight landing in France would be ’FR-1AA1020’.
8. Consider the set A = {a, b, c, d}.
(a) List all subsets of A with two elements. There are 6.
(b) List all ordered partitions of A of type (2, 1, 1). There are 12.
9. (a) The Mathletes are choosing club officers. If there are 22 members in the club, how many
ways can they select a President, Vice President, Secretary, and Treasurer?
(b) The Mathletes are choosing club officers. If there are 22 members in the club, 5 of whom
are Seniors. If a Senior must be president, how many ways can they select a President, Vice
President, Secretary, and Treasurer?
(c) Given that C(n, 3) = 680 and 3! = 6. Find P (n, 3).
(d) How many poker hands consist of 4 Aces and 1 Two? Recall a poker hand consists of exactly
5 cards.
(e) How many poker hands consist of A, King, Queen from one suit and two cards from a different
suit?
10. (a) A class of 20 students is going to play Jenga. The teacher will divide the class into groups.
One group will have 8 students, another group will have 5 students, and the last group will
have 7 students. In how many ways can the class be divided into these groups?
(b) At a training camp, 20 softball players will divide up to do different activities. 10 players will
play catch, 4 will practice hitting and 6 players will run. In how many ways can the players
be divided into these groups?
(c) At a national debate tournament, rules require that a team must consist of 3 men and two
women. There are 10 men and 4 women to choose from. In how many ways can the team be
formed?
(d) How many poker hands consist of at least 3 Clubs?
11. You are trying to make dinner and would like to use up some items in your fridge that are starting
to go bad. You have several onions that need to be used and some beef roast. Luckily, your
grandma just sent you a cookbook with 300 recipes.You look through your cookbook and make a
list of recipes that call for onion or beef roast. You have 100 total recipes on your list. You see
that 90 recipes call for onions and 30 call for beef roast.
(a) Draw a Venn diagram representing this situation. Fill in all appropriate numbers and make
sure to clearly label your diagram.
(b) How many recipes call for onions and not beef roast?
(c) How many recipes call for neither onions or beef roast?
(d) How many recipes call for onions and beef roast?
(e) How many recipes do not call for beef roast?
12. The following represents a map through a city. The lines represent streets (there are 7 streets
running north/south and 3 streets running east/west). All questions following refer to ”shortest
routes” from A to B. That is, only moves North and West are allowed.
B
C
A
(a) How many ”shortest routes” are there from A to B?
(b) How many ”shortest routes” do not have two consecutive Norths?
(c) How many ”shortest routes” pass through C?
13. The object is to start with the letter C on top and move down the diagram to a T at the bottom.
From any given letter, move only to one of the letters directly below it on the left or right. If
these rules are followed, how many different paths spell COUNT?
C
O
U
N
T
O
U
N
T
U
N
T
N
T
T
14. Does C(n, r) = C(n, n − r)? Explain why or give a counter example.
15. Does P (n, r) = P (n, n − r)? Explain why or give a counter example.
16. Calculate the following values.
(a) P (8, 2)
(d) C(8, 2)
(b) P (9, 8)
(e) C(9, 8)
(c) P (6, 3)
(f) C(6, 3)
17. Calculate the binomial and mulitnomial coefficients
5
(a)
3
(b)
5
2, 3
(c)
8
5, 3
(d)
10
2, 5, 3
(e)
5
2, 1, 1, 1