An element a in a ring R is called nilpotent if a" =0 for some positive integer n. The only nilpotent element in an integral domain is 0. Select one: O True False

The kindergartners and first and second graders were talking about their favorite field trips during the school year. One of the teachers, Ms. Brewer, turned the discussion into a math lesson, and the students conducted a survey. Each child wrote down on paper which trip was his or her favorite. (Note that many children named more than one field trip as their favorite.) The survey revealed that 55 wrote down the trip to the river, 48 indicated the trip to the police station, and 38 included the trip to the hardware store. The police station and the river were chosen on 32 papers, 26 papers included both the river and the hardware store, and 22 children wrote down the police station and the hardware store on their papers. Ms. Burke counted 17 papers that included all three and one that did not list any of the three trips. How many children were surveyed? How many children wrote down the river but not the police station? How many children did not list the hardware store?

For this problem, assume the test had 6 true-false questions and 8 multiple-choice questions, and that each multiple-choice question had 5 choices. How many different answer sheets can be submitted?

Find the number of different ways that an instructor can choose 7 students from a class of 14 students for a field trip.

Dr. Volz gave an exam with 14 questions, but students only had to answer 10 of them. Suppose the exam instructions say that either BOTH questions 1 and 2 may be included among the ten OR NEITHER is to be included.  How many different choices of 10 questions are there?

In the mid 1990 the state of Florida ran out of the combination of letters and numbers for license plates. Then, each license plate contained 3 letters of the alphabet followed by 3 digit from 0-10. How many combinations are there?