Learning Materials
Features
Discover
Chapter 1: Q.1.6 (page 15)
A well-known nursery rhyme starts as follows: 鈥淎s I was going to St. Ives I met a man with 7 wives. Each wife had 7 sacks. Each sack had 7 cats. Each cat had 7 kittens...鈥� How many kittens did the traveler meet
?
The number of kittens did the traveler meet is 2401 kittens
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
A student has to sell books from a collection of math, science, and economics books. How many choices are possible if
(a) both books are to be on the same subject?
(b) the books are to be on different subjects?
Consider the following combinatorial identity:
(a) Present a combinatorial argument for this identity by considering a set of people and determining, in two ways,
the number of possible selections of a committee of any size and a chairperson for the committee.
Hint:
(i) How many possible selections are there of a committee of size and its chairperson?
(ii) How many possible selections are there of a chairperson and the other committee members?
(b) Verify the following identity for :
localid="1648098528048"
For a combinatorial proof of the preceding, consider a set of n people and argue that both sides of the identity represent
the number of different selections of a committee, its chairperson, and its secretary (possibly the same as the chairperson).
Hint:
(i) How many different selections result in the committee containing exactly people?
(ii) How many different selections are there in which the chairperson and the secretary are the same?
(answer: .)
(iii) How many different selections result in the chairperson and the secretary being different?
(c) Now argue that
localid="1647960575612"
From a set of people, a committee of size is to be chosen, and from this committee, a subcommittee of size , , is also to be chosen.
(a) Derive a combinatorial identity by computing, in two ways, the number of possible choices of the committee and subcommittee鈥攆irst by supposing that the committee is chosen first and then the subcommittee is chosen, and second
by supposing that the subcommittee is chosen first and then the remaining members of the committee are chosen.
(b) Use part (a) to prove the following combinatorial identity:role="math" localid="1648189818817"
(c) Use part (a) and Theoretical Exercise 13 to show that:role="math" localid="1648189841030"
Consider -digit numbers where each digit is one of the integers . How many such numbers are there for which
(a) no two consecutive digits are equal?
(b) appears as a digit a total of times, ?
Suppose that fish are caught at a lake that contains distinct types of fish.
How many different outcomes are possible, where an outcome specifies the numbers of caught fish of each of the types?
How many outcomes are possible whenthe fish caught are trout?
How many when at least of the are trout?
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