91Ó°ÊÓ

A camera shop stocks eight different types of batteries. a. How many ways can a total inventory of 30 batteries be distributed among the eight different types? b. Assuming that one of the types of batteries is A76, how many ways can a total inventory of 30 batteries be distributed among the eight different types if the inventory must include at least four A76 batteries? c. If an inventory of 30 batteries is selected at random from the cight different types, what is the probability that at least four A76 batteries will be included? d. If an inventory of 30 batteries is selected at random from the eight different types, what is the probability that exactly four A 76 batteries will be included?

How many arrangements in a row of no more than three letters can be formed using the letters of the word NETWORK (with no repetitions allowed)?

$$ \text { Expand the expressions in 1-9 using the binomial theorem. } $$ $$ (u-v)^{5} $$

If \(n\) is a positive integer, how many 4-tuples of integers from 1 through \(n\) can be formed in which the elements of the 4-tuple are written in increasing order but are not necessarily distinct? In other words, how many 4-tuples of integers \((i, j, k, m)\) are there with \(1 \leq i \leq j \leq k \leq m \leq n ?\)

An urn contains 25 red balls and 15 blue balls. Two are chosen at random, one after the other, without replacement. a. What is the probability that both balls are red? b. What is the probability that the second ball is red but the first ball is not? c. What is the probability that the second ball is red? d. What is the probability that at least one of the balls is red?

a. How many five-digit integers (integers from 10,000 through 99,999 ) are divisible by 5 ? b. What is the probability that a five-digit integer chosen at random is divisible by \(5 ?\)

If \(n\) is a positive integer, how many 5 -tuples of integers from 1 through \(n\) can be formed in which the elements of the 5 -tuple are written in decreasing order but are not necessarily distinct? In other words, how many 5 -tuples of integers \((h, i, j, k, m)\) are there with \(n \geq h \geq i \geq j \geq k \geq m \geq 1\) ?

In a certain state, license plates consist of from zero to three letters followed by from zero to four digits, with the provision, however, that a blank plate is not allowed. a. How many different license plates can the state produce? b. Suppose 85 letter combinations are not allowed because of their potential for giving offense. How many different license plates can the state produce?

A student council consists of 15 students. a. In how many ways can a committee of six be selected from the membership of the council? b. Two council members have the same major and are not permitted to serve together on a committee. How many ways can a committee of six be selected from the membership of the council? c. Two council members always insist on serving on committees together. If they can't serve together, they won't serve at all. How many ways can a committee of six be selected from the council membership? d. Suppose the council contains eight men and seven women. (i) How many committees of six contain three men and three women? (ii) How many committees of six contain at least one woman? e. Suppose the council consists of three freshmen, four sophomores, three juniors, and five seniors. How many committees of eight contain two representatives from each class?

A computer programming team has 13 members. a. How many ways can a group of seven be chosen to work on a project? b. Suppose seven team members are women and six are men. (i) How many groups of seven can be chosen that contain four women and three men? (ii) How many groups of seven can be chosen that contain at least one man? (iii) How many groups of seven can be chosen that contain at most three women? c. Suppose two team members refuse to work together on projects. How many groups of seven can be chosen to work on a project? d. Suppose two team members insist on either working together or not at all on projects. How many groups of seven can be chosen to work on a project?