1.There arefivecandidates running for the presidential and vice-presidential nomination. How many ways are there of selecting presidential and vice-presidential candidates from the field? (Assume that every candidate is willing to accept either nomination.)

2.TheEnigma machinewas used by Germany in World War II to send coded messages. It has gained fame because it was an excellent coding device for its day and because of the ultimately successful efforts of the British (with considerable aid from the Poles) to crack the Enigma code. The breaking of the code involved, among other things, some very good mathematics developed by Alan Turing and others. One part of the machine consisted of three rotors, each containing theletters Athrough Z. To read an encrypted message, it was necessary to determine the initial settings of the three rotors (e.g., PDX or JJN). This is only the beginning of the problem of deciphering the Enigma code. Other parts of the machine allowed for many more initial settings. How many different initial settings of the three rotors are there?

The naval version of the Enigma machine had four rotors rather than three. How many initial settings were made possible by the rotors on the naval Enigma machine?

3.On December 1, 1969 (in the midst of the Vietnam War), the US Selective Service System conducted a lottery to determine the order in which young men(ages 18to 26) would be drafted into military service. The days of the year, includingFebruary 29,were assigned numbers from 1 to 366. Each such number was written on a slip of paper and placed in a plastic capsule. The366 capsuleswere placed in a large bowl. Then capsules were drawn, one at a time, from the bowl. The first number drawn corresponded toSeptember 14,and young men born on that day were the first group to be called up.

Assume that capsules were drawn at random. (This point has been the subject of much discussion.) What is the probability that a given young man would be in the first group to be called up? (Enter your probability as a fraction.)

4.On Halloween, a man presents a child with a bowl containingsixdifferent pieces of candy and six different pieces of gum. He tells her that she may have four of each. How many choices does she have?

5.Suppose there is a group of seven people from which we will make a committee.

In how many ways can we pick afive-person committee?

6.We want to select a committee of five members from a group ofeightwomen and six men. The order of selection is irrelevant.

What is the total number of committees we can make?

7.We want to select a committee of five members from a group offivewomen and six men. The order of selection is irrelevant.

How many committees can we make with Jack as committee chair? (Assume that only one member has that name!)

8.A dinner order at a restaurant consists of a salad with dressing, anentre,and three sides. The menu offers a salad with your choice ofranch, balsamic vinegar, or Italiandressing; a choice ofentrefromfried chicken, meat loaf, roast beef, catfish, or pork chops; and, as sides,fried okra, corn, black-eyed peas, green beans, French fries, rice, pinto beans, mashed potatoes, or squash. How many possible dinner orders are available at the restaurant? (Assume you must order three different sides.)