(1) Determine the generating function for the number of ways to distribute 35 pennies(from an unlimited supply) five children if: (a) Each child gets at least 2 cents? (b) The two youngest children...

(1) Determine the generating function for the number of ways to distribute 35 pennies(from an unlimited supply) five children if: (a) Each child gets at least 2 cents? (b) The two youngest children must get at least 10 cents? (2) The two cases of soft drinks 24 bottles of one type and 24 bottles of another, are distributed among five surveyors who are conducting taste tests. In how many ways can be the 48 bottles be distributed so that each surveyors gets: (a) At least two bottles of each? (b) At least two bottles of one particular type and at least three of the other? (3) How can Mary split up 12 hamburgers and 16 hot dogs among her son Richard, Peter, Christopher and James in such a way that James gets at least two hamburgers but at most five hot dogs? (4) Find generating function for the number of integer solution of : (a) 2w+3x+5y+7z=n 0=w,x,y,z (b) 2w+3x+5y+7z=n 0=w, 4=x,y, 5=z (5) How many 20 digits quaternary (0,1,2,3) sequence are there where: (a) There is at least one 2 and odd number of 0’s? (b) No symbol occurs exactly twice ? (c) There are exactly two 3’s or not at all?