How many permutations ofthe 10 digits either begin with

How many elements are in A I U A2 if there are 12 elements in A I , 18 elements in A2, and a) Al n A2 = 0? b) IA I n A2 1 = I? c) IAI n A2 1 = 6? d) Al S; A2?

There are 345 students at a college who have taken a course in calculus, 212 who have taken a course in discrete mathematics, and 188 who have taken courses in both calculus and discrete mathematics. How many students have taken a course in either calculus or discrete mathematics?

A survey of households in the United States reveals that 96% have at least one television set, 98% have telephone service, and 95% have telephone service and at least one television set. What percentage of households in the United States have neither telephone service nor a television set?

A marketing report concerning personal computers states that 650,000 owners will buy a modem for their machines next year and 1,250,000 will buy at least one software package. If the report states that 1,450,000 owners will buy either a modem or at least one software package, how many will buy both a modem and at least one software package?

Find the number of elements in A I U A2 U A3 ifthere are 100 elements in each set and if a) the sets are pairwise disjoint. b) there are 50 common elements in each pair of sets and no elements in all three sets. c) there are 50 common elements in each pair of sets and 25 elements in all three sets. d) the sets are equal.

Find the number of elements in A I U A2 U A3 if there are 100 elements in A I , 1000 in A2 , and 10,000 in A3 if a) Al S; A2 and A2 S; A3. b) the sets are pairwise disjoint. c) there are two elements common to each pair of sets and one element in all three sets.

There are 2504 computer science students at a school. Of these, 1876 have taken a course in Pascal, 999 have taken a course in Fortran, and 345 have taken a course in C. Further, 876 have taken courses in both Pascal and Fortran, 231 have taken courses in both Fortran and C, and 290 have taken courses in both Pascal and C. If 1 89 of these students have taken courses in Fortran, Pascal, and C, how many of these 2504 students have not taken a course in any of these three programming languages?

In a survey of270 college students, it is found that 64 like brussels sprouts, 94 like broccoli, 58 like cauliflower, 26 like both brussels sprouts and broccoli, 28 like both brussels sprouts and cauliflower, 22 like both broccoli and cauliflower, and 14 like all three vegetables. How many of the 270 students do not like any of these vegetables?

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, 3 12, and 344 students in these courses, respectively; 14 in both calculus and data structures; 213 in both calculus and programming languages; 21 1 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?

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

Find the number of positive integers not exceeding 100 that are either odd or the square of an integer.

Find the number of positive integers not exceeding 1000 that are either the square or the cube of an integer.

How many bit strings of length eight do not contain six consecutive Os?

How many permutations of the 26 letters of the English alphabet do not contain any ofthe strings fish, rat, or bird?

How many permutations ofthe 10 digits either begin with the 3 digits 987, contain the digits 45 in the fifth and sixth positions, or end with the 3 digits 123?

How many elements are in the union of four sets if each of the sets has 100 elements, each pair of the sets shares 50 elements, each three ofthe sets share 25 elements, and there are 5 elements in all four sets?

How many elements are in the union of four sets if the sets have 50, 60, 70, and 80 elements, respectively, each pair of the sets has 5 elements in common, each triple of the sets has 1 common element, and no element is in all four sets?

How many terms are there in the formula for the number of elements in the union of 10 sets given by the principle of inclusion-exclusion?

Write out the explicit formula given by the principle of inclusion-exclusion for the number of elements in the union of five sets.

How many elements are in the union of five sets if the sets contain 10,000 elements each, each pair of sets has 1000 common elements, each triple of sets has 100 common elements, every four of the sets have 10 common elements, and there is 1 element in all five sets?

Write out the explicit formula given by the principle of inclusion-exclusion for the number of elements in the union of six sets when it is known that no three ofthese sets have a common intersection.

Prove the principle of inclusion-exclusion using mathematical induction.

Let E 1, E2, and E3 be three events from a sample space S. Find a formula for the probability of E\ U E2 U E3 .

Find the probability that when a fair coin is flipped five times tails comes up exactly three times, the first and last flips come up tails, or the second and fourth flips come up heads.

Find the probability that when four numbers from 1 to 100, inclusive, are picked at random with no repetitions allowed, either all are odd, all are divisible by 3, or all are divisible by 5

Find a formula for the probability of the union of four events in a sample space if no three of them can occur at the same time.

Find a formula for the probability of the union of five events in a sample space if no four of them can occur at the same time.

Find a formula for the probability of the union of n events in a sample space when no two ofthese events can occur at the same time.

Find a formula for the probability of the union ofn events in a sample space.