Math 216 Extra Counting Examples
Dr. Allen - Spring 2017
1. A small business assigns account numbers for regular clients as follows. An account
number consists of six symbols. The first two symbols are taken from the letters A,B,C,
... , Z and the last four symbols are from the digits 1,2,3,4,5,6,7,8,9. The digit 0 is not
allowed as it is too often confused with the letter O. To illustrate, one possible account
number is AA2117.
a. How many different account numbers are possible?
b. How many different account numbers can be made by rearranging the symbols
AB3522, still using the two letters followed by 4 digits format?
2. A candy factory makes chocolate bars in three flavors (regular, dark, and milk
chocolate), four sizes, and with or without nuts. How many different types of chocolate
bar does this factory make?
3. In a twenty-question True/False test, each question can be answered with either a “T”
or an “F” or it can be left blank. (A wrong answer counts against the student, a blank is
just zero.)
a. In how many distinct ways can such a twenty-question test be completed?
b. How many distinct ways can such a test be completed if exactly 2 questions are left
blank, and the remaining have the same number of T's and F's?
c. How many distinct ways can such a test be completed if at most one question is left
blank?
4. A certain club consists of 12 husband-wife couples.
a. How many different five-person committees can be formed using these 24 people?
b. How many different five-person committees can be formed from these 24 people if
both an all male committee and an all female committee are not allowed?
c. If Nancy refuses to serve on a committee that includes either Barbara or Jason (or
both), how many different five-person committees can be formed?
5. Bobby has 12 identical blue plastic blocks and 12 identical red plastic blocks.
a. In how many different ways can Bobby arrange 12 of these blocks in a row?
b. In how many ways can Bobby arrange all 24 of these blocks in a row?
c. Bobby hides his blocks and three playmates, Alice, Greg, and Sarah, search to find
them. If all 24 blocks are found, in how many distinct ways can they end up distributed
among the three playmates?
d. Bobby took his 24 blocks to Billy's house. Upon returning home, Bobby found that
he had only 21 of his blocks. In how many different ways could Bobby have returned
with only 21 blocks?
6. A pizza restaurant offers four different sizes of pizza (personal, medium, large, or
party), three different types of crust (regular, deep dish, or thin), and can be ordered plain
(no toppings) or may have 1 or more of the following toppings: (1) extra cheese, (2)
pepperoni, (3) sausage, (4) beef, (5) Canadian bacon, (6) bbq chicken, (7) anchovies, (8)
pineapple, (9) mushrooms, (10) green pepper, (11) black olives, (12) peperoncini, (13)
artichokes, (14) tomato, (15) jalapeno, (16) onions, (17) feta cheese, and (18) ham. How
many different pizzas does the restaurant offer? [Examples: A medium deep dish pizza
with sausage, Canadian bacon, and black olives is one possible pizza. A personal pizza
with thin crust and no toppings is another possibility. A regular crust party pizza with
extra cheese is yet another choice.]
7. How many 6 letter words can be formed using the standard 26 letter alphabet if no
letter is allowed to be used more than once in any word? [A “word” is any six letters - it
need not be meaningful. For example, “abzmol” is a word.]
8. Linda has 60 different books. She wants to arrange these books on two shelves, 30
books to a shelf. All books are distinct and the only restriction is that she has a 3 volume
set of history books and she wants those to be together on one shelf and arranged in order
(volumes I, II, III arranged together and in order left to right). In how many ways can she
arrange these books?