91Ó°ÊÓ

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?

a) If we have \(k\) different colors available, in how many ways can we paint the walls of a pentagonal room if adjacent walls are to be painted with different colors? b) What is the smallest value of \(k\) for which such a coloring is possible?

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?

In how many ways can one arrange all of the letters in the word INFORMATION so that no pair of consecutive letters occurs more than once? [Here we want to count arrangements such as IINNOOFRMTA and FORTMAIINON but not INFORINMOTA (where "IN" occurs twice) or NORTFNOIAMI (where "NO" occurs twice).]

Determine the number of integer solutions to \(x_{1}+x_{2}+\) \(x_{3}+x_{4}=19\) where \(-5 \leq x_{i} \leq 10\) for all \(1 \leq i \leq 4\)

Determine the number of positive integers \(x\) where \(x \leq\) \(9.999,999\) and the sum of the digits in \(x\) equals 31 .

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?

Ms. Pezzulo teaches geometry and then biology to a class of 12 advanced students in a classroom that has only 12 desks. In how many ways can she assign the students to these desks so that (a) no student is seated at the same desk for both classes? (b) there are exactly six students each of whom occupies the same desk for both classes?

Give a combinatorial argument to verify that for all \(n \in \mathbf{Z}^{+}\), $$ n !=\left(\begin{array}{l} n \\ 0 \end{array}\right) d_{0}+\left(\begin{array}{l} n \\ 1 \end{array}\right) d_{1}+\left(\begin{array}{l} n \\ 2 \end{array}\right) d_{2}+\cdots+\left(\begin{array}{l} n \\ n \end{array}\right) d_{n}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) d_{k} $$ (For each \(1 \leq k \leq n, d_{k}=\) the number of derangements of 1 \(2,3, \ldots, k ; d_{0}=1 .\) )

Let \(D_{18}\) denote the set of positive divisors of \(18 .\) For \(d \in\) \(D_{18}\) let \(S_{d}=\\{n \mid 0