Question 917936: The Red Sox play the Yankees in a seven game series that ends when one team has won four games.
We record the outcome of a game with a W for a Red Sox win and an L for a Red Sox loss, e.g.
WWWW, WLWLWLW, or WWLLLWW.
i. How many possible outcomes are there?
ii. How many series would have to be played to be sure that the same outcome happens twice?
iii. If the Red Sox win in four games with a total of 17 runs, how many ways could their runs be
distributed among the four games? (Sample run distributions: (2, 7, 5, 3) (12, 1, 1, 3))
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! i. Casework on # games played (we only count # outcomes with 4 W's)
4 games: Only WWWW, 1 outcome
5 games: Any rearrangement of WWWWL with the last game W, 4C1 = 4 outcomes
6 games: Any rearrangement of WWWWLL, with the last game W, 5C2 = 10 outcomes
7 games: Any rearrangement of WWWWLLL, with the last game W, 6C3 = 20 outcomes
1+4+10+20 = 35. Multiply by 2 to account for outcomes with 4 L's --> 70 different outcomes
ii. 71 by Pigeonhole
iii. Equivalent to the number of 4-tuples whose sum is 17 and each element is at least 1 (since it is impossible to win with 0 runs). This is equal to (13+3)C3 = 16C3 = 560 (look up stars and bars if you're not familiar)
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