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Homework answers / question archive / 1) In how many ways can a group of five girls and two boys be made out of a total of seven boys and three girls? Fifteen planes are commuting between two cities A and B

1) In how many ways can a group of five girls and two boys be made out of a total of seven boys and three girls? Fifteen planes are commuting between two cities A and B. In how many ways can a person go from A to B and return by a different plane? Three cars enter a parking lot with 11 empty parking spaces. Each car has an option of backing into its parking space or driving in forward. How many ways can the three drivers select

i.Parking spaces?

ii.Parking spaces and parking positions?

2) In an election of the Student Government Association Executive Committee at a university, there are 13 student candidates, with 7 females and 6 males. Five persons are to be selected that include at least 3 females. In how many different ways can this committee be selected? In how many different ways can the letters of the word “MATHEMATICS” be arranged so that the vowels always come together?

3) Let A = {0,3,6,9} and B = {1,2,4,7,8}. Draw the Venn diagram illustrating these sets with the union as the universal set. Find the number of ways of choosing 2 consonants from 7 and 2 vowels from 4. A game team consists of ten players, six male and four female players. In how many different ways can four players be selected such that at least one male player is included in the selected group?

4) In a statistics course, students are to conduct a project and submit it at end of semester that they are enrolled in. The professor of the class has a list of 5 sets of possible projects, each containing 5, 10, 15, 20, and 15 topics different from each other, respectively. How many choices are there for each student to conduct a project?

5) The power set of Ω is a σ-algebra. From intervals (by means of the operations allowed in a σ-algebra), we can define a very important set, namely, the Borel set that we will discuss in Chapter 2. We will now try to define this set. Let us assume that a department of mathematics currently has 25 faculty members and 50 majors. One of the members of the Mathematics Grade Challenge Committee has resigned and needs to be replaced by either a faculty or a major. Thus, there are 25 + 50 = 75 choices for the mathematics department chairman to choose replacing the vacant position.

6) Consider a walker who walks on a real line starting at 0 with moving one step forward with probability *p *and backward with probability *q*, *pq*+=1. Let *Xn *describe the position of the walker after n steps. (a)What is the probability that walker is at the point 0 on the line after two steps? (b)What is the probability that walker is at the point −1 on the line after three steps? (c)What is the probability that walker is at the point 3 on the line after three steps? (d)Suppose the walker is at point 4 after 10 steps, does the probability that it will be at point 8 after 16 steps (6 more steps) depend on how it moves to point 4 within the first 10 steps? (e)Are *X*_{10}−*X*_{4} and *X*_{16}−*X*_{12} independent? (f)Are *X*_{10}−*X*_{4} and *X*_{12}−*X*_{8 }independent?