12.36. In this exercise, we formulate a more general version of the pigeonhole principle. (a) Let n1,n2,...,nk be positive integers. Show that if n pigeons with n E m + $12 + - - - + 12;; k + 1 are distributed among 1:: pigeonholes, then there exist a pigeonhole j with 1 s; j a; k that contains at least '11:; pigeons. (b) Derive the pigeonhole principle as a consequence of the generalized pigeon- hole principle given in part (a). (c) Suppose that you have a glass jar full of fasteners of three different types. You want to reattach a fence picket. You will either need two long screws of type 51 or three medium length screws of type 82 or four shOrt length screws of type 33. You want to attach the fence picket with just one type of screws for a uniform look. How man}r screws do you need to grab from the jar to guarantee that you have either two screws of type 81, three screws of type 32, or four screws of type 33