91Ó°ÊÓ

Determine the number of positive integers \(n, 1 \leq n \leq 2000\), that are a) not divisible by 2,3 , or \(5 .\) b) not divisible by \(2,3,5\), or 7 . c) not divisible by 2,3, or 5, but are divisible by \(7 .\)

a) List all the derangements of \(1,2,3,4,5\) where the first three numbers are 1,2 , and 3 , in some order. b) List all the derangements of \(1,2,3,4,5,6\) where the first three numbers are 1,2 , and 3 , in some order.

a) Find the rook polynomial for the standard \(8 \times 8\) chessboard. b) Answer part (a) with 8 replaced by \(n\), for \(n \in \mathbf{Z}^{+}\).

In how many ways can one arrange the letters in CORRESPONDENTS so that (a) there is no pair of consecutive identical letters? (b) there are exactly two pairs of consecutive identical letters? (c) there are at least three pairs of consecutive identical etters?

How many permutations of \(1,2,3,4,5,6,7\) are not derangements?

In how many ways can one distribute ten distinct prizes among four students with exactly two students getting nothing? How many ways have at least two students getting nothing?

Zelma is having a luncheon for herself and nine of the women in her tennis league. On the morning of the luncheon she places name cards at the ten places at her table and then leaves to run a last-minute errand. Her husband, Herbert, comes home from his morning tennis match and unfortunately leaves the back door open. A gust of wind scatters the ten name cards. In how many ways can Herbert replace the ten cards at the places at the table so that exactly four of the ten women will be seated where Zelma had wanted them? In how many ways will at least four of them be seated where they were supposed to be?

How many derangements of \(1,2,3,4,5,6,7,8\) start with (a) \(1,2,3\), and 4 , in some order? (b) \(5,6,7\), and 8 , in some order?

Professor Bailey has just completed writing the final examination for his course in advanced engineering mathematics. This examination has 12 questions, whose total value is to be 200 points. In how many ways can Professor Bailey assign the 200 points if (a) each question must count for at least 10 , but no more than 25 , points? (b) each question must count for at least 10 , but not more than 25 , points and the point value for each question is to be a multiple of 5 ?

At Flo's Flower Shop, Flo wants to arrange 15 different plants on five shelves for a window display. In how many ways can she arrange them so that each shelf has at least one, but no more than four, plants?