91Ó°ÊÓ

(a) In how many ways can the months of the birthdays of five people be distinct? (b) How many possibilities are there for the months of the birthdays of five people? (c) In how many ways can at least two people among five have their birthdays in the same month?

Refer to a set of five distinct computer science books, three distinct mathematics books, and two distinct art books. In how many ways can these books be arranged on a shelf if all five computer science books are on the left and both art books are on the right?

Show that there are \((2 n-1)(2 n-3) \cdots 3 \cdot 1\) ways to pick \(n\) pairs from \(2 n\) distinct items.

Refer to a set of five distinct computer science books, three distinct mathematics books, and two distinct art books. In how many ways can these books be arranged on a shelf if the two art books are not together?

Show that the number of ways that \(2 n\) persons, seated around a circular table, can shake hands in pairs without any arms crossing is \(C_{n}\), the \(n\) th Catalan number.

How many antisymmetric relations are there on an \(n\) -element set?

How many symmetric and antisymmetric relations are there on an \(n\) -element set?

How many reflexive, symmetric, and antisymmetric relations are there on an \(n\) -element set?

A six-person committee composed of Alice, Ben, Connie, Dolph, Egbert, and Francisco is to select a chairperson, secretary, and treasurer. How many selections are there in which either Connie is chairperson or Alice is an officer or both?

Prove the Inclusion-Exclusion Principle for three finite sets: $$|X \cup Y \cup Z|=|X|+|Y|+|Z|-|X \cap Y|-|X \cap Z|-|Y \cap Z|+|X \cap Y \cap Z|.$$