Question 1 Solve the following counting problems: (a) How many license plates can be made by using 3 letters following by 4 digits? (b) How many license plates can be made by using either 3 letters or 4 digits? Question 2 Suppose there are 10 distinguish balls in a bag, where 5 balls are red, 3 are blue and 2 are yellow. Solve the following problems: (a) How many ways are there to take 4 balls out of the bag (the order is not considered)? (b) How many ways are there to take 4 balls out of the bag such that all of the balls are red (the order is not considered)? (c) What is the probability that the 4 balls chosen from the bag are all red? Write the experiment, sample space, and event (Hint: use the results of (a) and (b)) Question 3 Ro two dices. What is the probability that the sum of two dices is 6? Write the experiment, sample space, and event Question 4 Let A 2, 4, 6, 8 and B fa, b, c. Consider R 2.c),(4,b). (6,a). (2.d), (8,a), R'-((a,2), (c,4), (6,b), (8,b) and R" (2,a), (4,c), (8,b), (6,a), and Answer the following questions: (a) Among R, R' and R" which are the relations from A to B? (b) Among R, R' and R" which are the functions from A to B? Question 5 Consider the following relations on the set of integers. R, a,b) lasb R. = {(a, b) | a b) R a,b) a b R-a,b) a mod(3) bmod(3) R, a,b) a b) (a) Which relations contain the pair (3, 6)? b) Which relations are reflexive? (c Which relations are symmetric? d) Which relations are antisymmetric? e Which relations are transitive