3. a. In how many ways can the 12 members of the Board of Directors (BOD) be chosen from 12 parent-nominees and seven teacher-nominees if there must be eight parents in the BOD? b. After the 12 members are chosen, in how many ways can they elect among themselves the seven top positions (president, vice president, and others)?

85 cities were surveyed to determine sports teams. 25 had soccer, 19 had rugby, 18 had basketball, 11 had soccer and rugby, 12 had soccer and basketball, 11 had rugby and basketball. 6 had all three. Let A = soccer, B = rugby, C = basketball. How many had only a basketball team? 1 How many had soccer and rugby, but not basketball? 5 How many had soccer or rugby? 33 How many had soccer or rugby, but not basketball? 16 How many had exactly two teams? U A I IV II VII VI III с B VIII

7. A club has 10 seniors, 12 juniors, 8 sophomores, and 6 freshman. How many ways can the club choose a 6-person party-planning committee with exactly 4 juniors?

1. Assume that a class of 16 students is supposed to write an essay. If 3 essays are randomly selected to appear on the class bulletin board, how many different groups of 3are possible? (calculate number of combinations of 3 items from 16 available items).Show the result of calculations.

A corporation must appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO). It must also appoint a planning committee with three different members. There are 16 qualified candidates, and officers can also serve on the committee. Complete parts (a) through (c) below. a. How many different ways can the officers be appointed? There are different ways to appoint the officers. b. How many different ways can the committee be appointed? There are different ways to appoint the committee. c. What is the probability of randomly selecting the committee members and getting the three youngest of the qualified candidates? P(getting the three youngest of the qualified candidates) = (Type an integer or a simplified fraction.)

2. How many ways are there to place 12 people into four groups of equal size?