A certainbaseball team has 19 players. Only nine can be on the field ata time. Each of the nine players on the field has a distinct fieldposition: pitcher, catcher, first baseman, second baseman, thirdbaseman, short stop, left field, right field, or center field.Assume for the moment that every player is qualified to play everyposition.

a) How may ways are there to select nine of the 19players to be on the field (without regard to which position eachplayer will have)?

b) How many ways are there to assign field positionsto a given set of nine players?

c) How many ways are there to fill the nine fieldpositions from among the 19 players (taking account of the positioneach player will have)?

d) How many ways are there to fill either the pitcheror catcher field position (but not both) from among the 19 players(leaving the other field positions empty)?

Now supposethat not every player can play in every position. The outfielders(left field, center field, right field) can play any outfieldposition, the infielders (1st base, 2nd base, 3rd base, short stop)can play any infield position, the pitchers can only pitch, and thecatchers can only catch. Suppose a certain team has 20players, of whom 3 are catchers, 4 are outfielders, 6 areinfielders, and 7 are pitchers.

e) How many ways can the team assign field positionsto 9 of the 20 players, putting each of the 9 selected players in aposition he can play, and ensuring that all 9 field positions arefilled?