466 CHAPTER 7 Counting and Combinatorics
- How many ways can you choose eight letters from
aaaaa bbbbbb ccccccccwith at least one a, one b, and two c's?- How many integer solutions are there for the following equations?
(a) x+y+z=8wherex>0, y>0, andz>0
(b) x + y + z + t = 18 where x, y, z, and t are each greater than zero
(c) x+y+z+t=12wherex>1,y>_2, andt > 1
- How many ways can two booksellers divide between themselves 300 copies of one
book, 200 copies of another, and 100 copies of a third if neither bookseller is to get all
the copies of any one of the books? - A bookbinder is to bind 10 different books using red, green, or blue cloth for each
book. How many ways can this be done if each color is to be used for:
(a) At least one book
(b) At least two books
(c) With no restriction on the number of times a color may be used - How many ways can six candy bars be distributed among three children if every child
is to receive at least one candy bar? - How many ways can a class of 25 students be assigned to three different lab sections
if each lab section has at least 5 students? - A king is placed on the bottom left-hand square of an 8 x 8 chess board and is to move
to the top right-hand comer square. If the piece can move only up or to the right, how
many possible paths does it have? - An XYZ-3000 is a front-end processor to five mainframe computers at RST U. There
are 64 incoming phone lines to the XYZ-3000. In how many ways can the front-end
processor assign lines to computers so that 8 are directed to C 1 , 14 to C 2 , 17 to C 3 , 16
to C 4 , and the remaining to C 5? A program called TUNE monitors the performance
of a computer system. Suppose each user is assigned to one of the 64 memory areas
when first logged onto the system. TUNE samples memory areas or partitions when
a user is first logged on to the system to decide how to assign the new user memory.
How many can this be done if TUNE samples 17 of 64 system partitions? How many
if the one fixed partition Z is always excluded from the sample? How many if two
fixed partitions are always chosen? - Three first-year, three second-year, and three third-year students are to be seated in
a row. The students in each class are indistinguishable. How many ways can they be
seated so that no three students of the same class sit together? - How many ways can an examiner assign 90 points to 12 questions with each question
getting at least four points? - A shop sells six flavors of ice cream. Each ice cream cone holds one, two, or three
scoops of ice cream. How many ways can four ice cream cones be made such that:
(a) All the cones have a different flavor, and each cone has a single flavor for each of
its scoops.
(b) Not necessarily all the cones have a different flavor.
(c) The cones contain only two or three flavors of ice cream.
(d) The cones contain three different flavors.