91Ó°ÊÓ

Determine how many \(n \in \mathbf{Z}^{+}\)satisfy \(n \leq 500\) and are not divisible by \(2,3,5,6,8\), or 10 .

a) In how many ways can the letters in ARRANGEMENT be arranged so that there are exactly two pairs of consecutive identical letters? at least two pairs of consecutive identical letters? b) Answer part (a), replacing two with three.

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 \(n \in \mathbf{Z}^{+}\).

How many derangements are there for \(1,2,3,4,5 ?\)

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 letters?

Find the number of positive integers \(n\) where \(1 \leq n \leq 1000\) and \(n\) is not a perfect square, cube, or fourth power.

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

a) Let \(A=\\{1,2,3, \ldots, 7\\}\). A function \(f: A \rightarrow A\) is said to have a fixed point if for some \(x \in A, f(x)=x\). How many one-to-one functions \(f: A \rightarrow A\) have at least one fixed point? b) In how many ways can we devise a secret code by assigning to each letter of the alphabet a different letter to represent it?

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