Of the four teams in a softball league, one team has four pitchers and the other

Question

Of the four teams in a softball league, one team has four pitchers and the other teams have three each. Determine each of the following.

     (a) The number of possible selections of pitchers for an all-star team, if
           exactly four pitchers are to be chosen.

     (b) The number of possible selections if one pitcher is to be chosen from
           each team.

     (c) The number of possible selections of four pitchers, if exactly two of the
           five left-handed pitchers in the league must be selected.

     (d) The number of possible orders in which the four pitchers, once they are
           selected, can appear (one at a time) in the all-star game.

Explanation / Answer

a) Here we just select 4 out of the 13 total pitchers, without replacement, and order doesn't matter

So the answer is 13C4 = 13!/(9!)(4!) = 715

b) Here we use the rule of AND to select a pitcher from team 1 AND one from team 2, AND one from team 3 And one from team 4. So we multiply the number of ways to get an answer of 4 * 3* 3 * 3 = 108

c) Now we choose 2 out of 5 lefthanded pitchers AND 2 out of 8 righties.

So we multiply 5C2 * 8C5 = 10 * 56 = 560

d) Once we have the 4 pitchers, we have 4 choices for who pitches first, AND 3 choices of who pitches second AND2 choices for the third pitcher and AND 1 choice for the last pitcher. (This is done without replacement, but order matters). So the answer is 4! = 4*3*2*1 = 24

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