91Ó°ÊÓ

Five people are to be seated around a circular table. Two seatings are considered the same if one is a rotation of the other. How many different seatings are possible?

A non-leap year has 365 days. Assume that January 1 is a Monday. a. How many Sundays are there in the year? b. How many Mondays are there in the year?

$$ \text { Write all the } 3 \text {-permutations of }\\{s, t, u, v\\} \text {. } $$

Prove the inclusion/exclusion rule for two sets \(A\) and \(B\) by showing that \(A \cup B\) can be partitioned into \(A \cap B\), \(A-(A \cap B)\), and \(B-(A \cap B)\), and then using the addition and difference rules.

Prove the inclusion/exclusion rule for three sets.

a. How many 3-permutations are there of a set of five objects? b. How many 2-permutations are there of a set of eight objects?

Use mathematical induction to prove the general inclusion/exclusion rule: If \(A_{1}, A_{2}, \ldots, A_{n}\) are finite sets, then $$ \begin{aligned} N\left(A_{1} \cup A_{2} \cup \ldots \cup A_{n}\right) & \sum_{1 \leq i \leq n} N\left(A_{i}\right)-\sum_{1 \leq i=j \leq n} N\left(A_{i} \cap A_{j}\right) \\\ &+\sum_{1 \leq j

a. How many ways can three of the letters of the word \(A L G O R I T H M\) be selected and written in a row? b. How many ways can six of the letters of the word ALGORITHM be selected and written in a row? c. How many ways can six of the letters of the word ALGORITHM be selected and written in a row if the first letter must be \(A\) ? d. How many ways can six of the letters of the word ALGORITHM be selected and written in a row if the first two letters must be \(O R\) ?