91Ó°ÊÓ

There are 2504 computer science students at a school. Of these, 1876 have taken a course in Java, 999 have taken a course in Linux, and 345 have taken a course in C. Further, 876 have taken courses in both Java and Linux, 231 have taken courses in both Linux and C, and 290 have taken courses in both Java and C. If 189 of these students have taken courses in Linux, Java, and C, how many of these 2504 students have not taken a course in any of these three programming languages?

a) Find a recurrence relation for the number of bit strings of length n that contain a pair of consecutive 0s. b) What are the initial conditions? c) How many bit strings of length seven contain two consecutive 0s?

In how many ways can a \(2 \times n\) rectangular checkerboard be tiled using \(1 \times 2\) and \(2 \times 2\) pieces?

How many positive integers less than \(10,000\) are not the second or higher power of an integer?

a) Find a recurrence relation for the number of bit strings of length n that contain three consecutive 0s. b) What are the initial conditions? c) How many bit strings of length seven contain three consecutive 0s?

How many onto functions are there from a set with seven elements to one with five elements?

a) Find a recurrence relation for the number of bit strings of length n that do not contain three consecutive 0s. b) What are the initial conditions? c) How many bit strings of length seven do not contain three consecutive 0s?

How many students are enrolled in a course either in calculus, discrete mathematics, data structures, or programming languages at a school if there are 507, 292, 312, and 344 students in these courses, respectively; 14 in both calculus and data structures; 213 in both calculus and programming languages; 211 in both discrete mathematics and data structures; 43 in both discrete mathematics and programming languages; and no student may take calculus and discrete mathematics, or data structures and programming languages, concurrently?

How many ways are there to distribute six different toys to three different children such that each child gets at least one toy?

Find the number of positive integers not exceeding 100 that are not divisible by 5 or by 7 .