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Chapter 1: Q. 1.2 (page 19)
If Americans, French people, and British people are to be seated in a row, how many seating arrangements are possible when people of the same nationality must sit next to each other?
The required no. of seating arrangements are.
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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).
The following identity is known as Fermat鈥檚 combinatorial identity:
Give a combinatorial argument (no computations are needed) to establish this identity.
Hint: Consider the set of numbers through . How many subsets of size have as their highest numbered member?
Prove the generalized version of the basic counting principle.
In Problem 21, how many different paths are there from A to B that go through the point circled in the following lattice?
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).
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