Consider a set A with 10 elements.  How many different functions are there from the set A to a set with three elements?

Teachers get courses assigned to teach each semester. For each instructor, there are the courses that the instructor can teach based on the skill set of the instructor, and there are courses that the teacher would rather teach all the time, closer to their specialization. To be able to teach in any department, a teacher must be able to teach more than the favorite courses. Let X denote the proportion of teachers who teach the whole spectrum of courses taught in a department, and Y the proportion of teachers who teach the courses they specialize in. Let X and Y have the joint density function : f(x,y)=2(x+y),0<y<x<1 (a) Given that 10% of the teachers teach the whole spectrum of courses, what is the probability that fewer than 5% teach their favorite courses? (b)  What is the expected percentage of teachers teaching their favorite courses when the proportion of teaching the whole spectrum is 0.7?