Problem 4. In a state lottery, four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select (1) 6, 7, 8, 9 (2) 7, 7, 8, 8 (3) 7, 8, 8, 8 Problem 5. (1) How many subsets of { 1, 2, 3, 4, 5, 6, 7, 8} contain at least one of the elements of {1, 2, 3}? (2) How many subsets of {1, 2, 3, 4, 5, 6, 7, 8} of size two (two elements) contain at least one of the elements of {1, 2, 3}? Problem 6. (1) Using the binomial expansion theorem we discussed in the class, show that n [(-1) (") = 0. r=0 (2) Using the identy in part (a), argue that the number of subsets of a set with n elements that contain an even number of elements is the same as the number of subsets that contain an odd number of elements