To test the effects of a new drug, you plan the following clinical trial: Each patient

Suppose that you want to investigate the influence of light and fertilizer levels on plant performance. You plan to use five fertilizer and two light levels. For each combination of fertilizer and light level, you want four replicates.What is the total number of replicates?

Suppose that you want to investigate the effects of leaf damage on the performance of drought-stressed plants. You plan to use three levels of leaf damage and four different watering protocols. For each combination of leaf damage and watering protocol, you plan to have three replicates. What is the total number of replicates?

, a lady beetle, is an important predator of egg masses of , the European corn borer. also feeds on aphids and maize pollen. To study its food preferences, you choose two satiation levels for and combinations of two of the three food sources (i.e., either egg masses and aphids, egg masses and pollen, or aphids and pollen). For each experimental protocol, you want 20 replicates. What is the total number of replicates?

To test the effects of a new drug, you plan the following clinical trial: Each patient receives the new drug, an established drug, or a placebo. You enroll 50 patients. In how many ways can you assign them to the three treatments?

The Muesli-Mix is a popular breakfast hangout near a campus. A typical breakfast there consists of one beverage, one bowl of cereal, and a piece of fruit. If you can choose among three different beverages, seven different cereals, and four different types of fruit, how many choices for breakfast do you have?

To study sex differences in food preferences in rats, you offer one of three choices of food to each rat. You plan to have 12 rats for each food-and-sex combination.Howmany rats will you need?

The genome of the HIV virus consists of 9749 nucleotides. There are four different types of nucleotides. Determine the total number of different genomes of size 9749 nucleotides.

Automated chemical synthesis of DNA has made it possible to custom-order moderate-length DNA sequences from commercial suppliers. Assume that a single nucleotides weighs about 5.6 1022 gram and that there are four kinds of nucleotides. If you wish to order all possibleDNAsequences of a fixed length, at what length will your order exceed (a) 100 kg and (b) the mass of the Earth (5.9736 1024 kg)?

You plan a trip to Europe during which you wish to visit London, Paris, Amsterdam, Rome, and Heidelberg. Because you want to buy a railway ticket before you leave, you must decide on the order in which you will visit these five cities. How many different routes are there?

Five people line up for a photograph. How many different lineups are possible?

You have just bought seven different books. In how many ways can they be arranged on your bookshelf?

Four cars arrive simultaneously at an intersection. Only one car can go through at a time. In how many different ways can they leave the intersection?

How many four-letter words with no repeated letters can you form from the 26 letters of the alphabet?

A committee of 3 people must be chosen from a group of 10. The committee consists of a president, a vice president, and a treasurer. How many committees can be selected?

Three different awards are to be given to a class of 15 students. Each student can receive at most one award. Count the number of ways these awards can be given out.

You have just enough time to play 4 songs out of 10 from your favorite CD. In how many ways can you program your CD player to play the 4 songs?

Six customers arrive at a bank at the same time. Only one customer at a time can be served. In how many ways can the six customers be served?

An amino acid is encoded by triplet nucleotides. How many different amino acids are possible if there are four different nucleotides that can be chosen for a triplet?

A bag contains 10 different candy bars. You are allowed to choose 3. How many choices do you have?

During International MovieWeek, 60 movies are shown. You have time to see 5 movies. How many different plans can you make?

A committee of 3 people must be formed from a group of 10. How many committees can there be if no specific tasks are assigned to the members?

A standard deck contains 52 different cards. In how many ways can you select 5 cards from the deck?

An urn contains 15 different balls. In how many ways can you select 4 balls without replacement?

Twelve people wait in front of an elevator that has room for only 5. Count the number of ways that the first group of people to take the elevator can be chosen.

Four As and five Bs are to be arranged into a nine-letter word. How many different words can you form?

Suppose that you want to plant a flower bed with four different plants. You can choose from among eight plants. How may different choices do you have?

Amin owns a 4-GB music storage device that holds 1000 songs. How many different playlists of 20 songs are there if the order of the songs is important?

Abookstore has 300 science fiction books. Molly wants to buy 5 of the 300 science fiction books. How many selections are there?

A box contains five red and four blue balls. You choose two balls. (a) How many possible selections contain exactly two red balls, how many exactly two blue balls, and how many exactly one of each color? (b) Show that the sum of the number of choices for the three cases in (a) is equal to the number of ways that you can select two balls out of the nine balls in the box.

Twelve children are divided up into three groups, of five, four, and three children, respectively. In how many ways can this be done if the order within each group is not important?

Five As, three Bs, and six Cs are to be arranged into a 14- letter word. How many different words can you form?

A bag contains 45 beans of three different varieties. Each variety is represented 15 times in the bag. You grab 9 beans out of the bag. (a) Count the number of ways that each variety can be represented exactly three times in your sample. (b) Count the number of ways that only one variety appears in your sample.

Let S = {a, b, c}. List all possible subsets, and argue that the total number of subsets is 23 = 8.

Suppose that a set contains n elements. Argue that the total number of subsets of this set is 2n.

In how many ways can Brian, Hilary, Peter, and Melissa sit on a bench if Peter and Melissa want to be next to each other?

Paula, Cindy, Gloria, and Jenny have dinner at a round table. In how many ways can they sit around the table if Cindy wants to sit to the left of Paula?

In how many ways can you form a committee of three people from a group of seven if two of the people do not want to serve together?

In how many ways can you form two committees of three people each from a group of nine if (a) no person is allowed to serve on more than one committee? (b) people can serve on both committees simultaneously?

A collection contains seeds for four different annual and three different perennial plants. You plan a garden bed with three different plants, and you want to include at least one perennial. How many different selections can you make?

In diploid organisms, chromosomes appear in pairs in the nuclei of all cells except gametes (sperm or ovum). Gametes are formed during meiosis, a process in which the number of chromosomes in the nucleus is halved; that is, only one member of each pair of chromosomes ends up in a gamete. Humans have 23 pairs of chromosomes. How may kinds of gametes can a human produce?

Sixty patients are enrolled in a small clinical trial to test the efficacy of a new drug against a placebo and the currently used drug. The patients are divided into 3 groups of 20 each. Each group is assigned one of the three treatments. In how many ways can the patients be assigned?

One hundred patients wish to enroll in a small study in which patients are divided into four groups of 25 patients each. In how many ways can this be done if no patient is to be assigned to more than one group?

Expand (x + y)4.

Expand (2x 3y)5.

In how many ways can four red and five black cards be selected from a standard deck of cards if cards are drawn without replacement?

In how many ways can two aces and three kings be selected from a standard deck of cards if cards are drawn without replacement?

In the game of poker, determine the number of ways exactly two pairs can be picked.

In the game of poker, determine the number of ways a (five cards of the same suit) can be picked.

In the game of poker, determine the number of ways (four cards of the same value, plus one other cards) can be picked.

In the game of poker, determine the number of ways a (five cards with consecutive values, such as A 2 3 4 5 or 7 8 9 10 J or 10 J Q K A, but not all of the same suit) can be picked.

Counterpoint is a musical term that means the combination of simultaneous voices; it is synonymous with . In , three voices are arranged such that any voice can take any place of the three possible positions: highest, intermediate, and lowest voice. In how many ways can the three voices be arranged?

Counterpoint is a musical term that means the combination of simultaneous voices; it is synonymous with . In , five voices are arranged such that any voice can take any place of the five possible positions: from highest to lowest voice. In how many ways can the five voices be arranged?