2. Solve the following counting problems using the counting principles (Addition, Mul- tiplication, Inclusion and Exclusion, Pigeonhole), and/or Permutations/Combinations (a) A president and vice-president must be chosen for the executive committee of an organization. There are 17 volunteers from the Eastern Division and 24 volunteers from the Western Division. If both officers must come from the same division, in how many ways can the officers be selected? (b) After serving 137 customers, a cafeteria notes at the end of the day that 56 orders of green beans were sold, 38 orders of beets were sold, and 17 customers purchased th green beans and beets. How many customers bought neither beans nor beets? (c) Let n be a positive integer. Show that in any set of n +1 positive integers, there bo are at least two with the same remainder when divided by n (d) A hostess wishes to invite 6 dinner guests from a list of 14 friends. In how many ways can she choose her guests if two of her friends dislike each other and neither will come if the other is present? (e) A cheese shop carries a large stock of 34 kinds of cheese. By the end of the day, 48 cheese sales have been made, and the items sold must be restocked. How many different restocking orders are possible