Learning Materials
Features
Discover
Chapter 1: Q. 1.15 (page 16)
A dance class consists of students, of which are women and 12 are men. If men and women are to be
chosen and then paired off, how many results are possible?
The possible results are.
Over 30 million students worldwide already upgrade their learning with 91影视!
All the tools & learning materials you need for study success - in one app.
Get started for free
From a group of people, suppose that we want to choose a committee of k, , one of whom is to be designated as chairperson.
(a) By focusing first on the choice of the committee and then on the choice of the chair, argue that there are role="math" localid="1647945358534" possible choices.
(b) By focusing first on the choice of the non-chair committee members and then on the choice of the chair, argue that there are role="math" localid="1647945372759" possible choices.
(c) By focusing first on the choice of the chair and then on the choice of the other committee members, argue that
there are role="math" localid="1647945385288" possible choices.
(d) Conclude from parts (a), (b), and (c) that role="math" localid="1647945400273" .
(e) Use the factorial definition of to verify the identity in part (d).
In how many ways can identical balls be distributed into urns so that the urn contains at least balls, for each ? Assume that .
Give an analytic verification of
Now, give a combinatorial argument for this identity.
Consider a tournament of contestants in which the outcome is an ordering of these contestants, with ties allowed. That is, the outcome partitions the players into groups, with the first group consisting of the players who tied for first place, the next group being those who tied for the next-best position, and so on. Let localid="1648231792067" denote the number of different possible outcomes. For instance, localid="1648231796484" , since, in a tournament with localid="1648231802600" contestants, player localid="1648231807229" could be uniquely first, player localid="1648231812796" could be uniquely first, or they could tie for first.
(a) List all the possible outcomes when .
(b) With localid="1648231819245" defined to equal localid="1648231826690" , argue without any computations, that localid="1648281124813"
Hint: How many outcomes are there in which localid="1648231837145" players tie for last place?
(c) Show that the formula of part (b) is equivalent to the following:
localid="1648285265701"
(d) Use the recursion to find N(3) and N(4).
How many different letter arrangements can be made from the letters (a) Fluke? (b) Propose? (c) Mississippi? (d) Arrange?
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine