Three different classes contain 20, 18, and 25 students, respectively, and no student is a member of more than one class. If a team is to be composed of one student from each of these three classes, in how many different ways can the members of the team be chosen?

A college offers 2 introductory courses in​ history, 1 in​ science, 1in​ mathematics, 4 in​ philosophy, and 2 in English.   a. If a freshman takes one course in each area during her first​ semester, how many course selections are​ possible?       b. If a​ part-time student can afford to take only one introductory​ course, how many selections are​ possible?

There are 53 students.  Of these, 25 are taking Business, 26 are taking English, and 25 are taking Math.  14 are taking both Business and Math, 13 are taking Math and English, and 17 are taking Business and English.  12 students are taking none of these three classes.  How many students are taking Business but are not taking Math?

Four members from a 60​-person committee are to be selected randomly to serve as​ chairperson, vice-chairperson,​ secretary, and treasurer. The first person selected is the​ chairperson; the​ second, the​ vice-chairperson; the​ third, the​ secretary; and the​ fourth, the treasurer. How many different leadership structures are​ possible?

In a group of 35 music students, 16 students play an instrument, 16 students sing, and 19 students dance. Furthermore, 5 students both play and sing, 7 students both sing and dance, 6 students both play and dance, and 2 students do all three. How many students are engaged in neither of the three activities?

If there are 11 employees in a club, in how many ways can they choose a dishwasher and a bouncer?

Three different classes contain 20, 18, and 25 students, respectively, and no student is a member of more than one class. If a team is to be composed of one student from each of these three classes, in how many different ways can the members of the team be chosen?

Four members from a 36​-person committee are to be selected randomly to serve as​ chairperson, vice-chairperson,​ secretary, and treasurer. The first person selected is the​ chairperson; the​ second, the​ vice-chairperson; the​ third, the​ secretary; and the​ fourth, the treasurer. How many different leadership structures are​ possible?

A company is organizing a team-building event with various activities. There are 12 employees attending the event, and they need to be split into three teams: Team A, Team B, and Team C. Each team should have at least 2 members but no more than 6 members. In how many different ways can the teams be formed? (Note: The order of the teams doesn't matter; i.e., Team A, Team B, and Team C is the same as Team C, Team A, and Team B.)

From a club of 19 ​people, in how many ways can a group of four members be selected to attend a​ conference?