91Ó°ÊÓ

Prove that for any positive integer \(n,\) there exists a positive integer which, when expressed in decimal, consists of at most \(n\) Os and 1 s and is a multiple of \(n .\) Hint: Consider the set of the \(n\) integers, \(\\{1,11,111, \ldots\\},\) using only \(1 \mathrm{~s},\) and the remainders of these numbers when divided by \(n\). Answer Exercises \(18-21\) to give an argument that shows that if \(X\) is any \((n+2)\) -element subset of \(\\{1,2, \ldots, 2 n+1\\}\) and \(m\) is the greatest element in \(X,\) there exist distinct \(i\) and \(j\) in \(X\) with \(m=i+j .\) For each element \(k \in X-\\{m\\}\), let $$ a_{k}=\left\\{\begin{array}{ll} k & \text { if } k \leq \frac{m}{2} \\ m-k & \text { if } k>\frac{m}{2}. \end{array}\right. $$

Exercises \(10-19\) refer to dice that are loaded so that the numbers 2, \(4,\) and 6 are equally likely to appear. \(1,3,\) and 5 are also equally likely to appear, but 1 is three times as likely as 2 is to appear. Two dice are rolled. What is the probability of getting a sum of 6 given that at least one die shows \(2 ?\)

In how many ways can 10 distinct books be divided among three students if the first student gets five books, the second three books, and the third two books?

Exercises \(10-19\) refer to dice that are loaded so that the numbers 2, \(4,\) and 6 are equally likely to appear. \(1,3,\) and 5 are also equally likely to appear, but 1 is three times as likely as 2 is to appear. Two dice are rolled. What is the probability of getting a sum of 6 or doubles given that at least one die shows \(2 ?\)

In how many ways can five distinct Martians and eight distinct Jovians wait in line if no two Martians stand together?

Give a combinatorial argument to show that $$C(n, k)=C(n, n-k)$$

Four microprocessors are randomly selected from a lot of 100 microprocessors among which 10 are defective. Find the probability of obtaining no defective microprocessors.

Four microprocessors are randomly selected from a lot of 100 microprocessors among which 10 are defective. Find the probability of obtaining exactly one defective microprocessor.

In the California Daily 3 game, a contestant must select three numbers among 0 to 9 , repetitions allowed. A "straight play" win requires that the numbers be matched in the exact order in which they are randomly drawn by a lottery representative. What is the probability of choosing the winning numbers?

In the California Daily 3 game, a contestant must select three numbers among 0 to \(9 .\) One type of "box play" win requires that three numbers match in any order those randomly drawn by a lottery representative, repetitions allowed. What is the probability of choosing the winning numbers, assuming that the contestant chooses three distinct numbers?