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Search: id:A060853
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| A060853 |
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Number of possible games of 10-pin bowling with a total score of n. |
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+0
4
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| 1, 20, 210, 1540, 8855, 42504, 177100, 657800, 2220075, 6906900, 20030010, 54627084, 141116637, 347336412, 818558424, 1854631380, 4053948342, 8574134256, 17590903116, 35084425512, 68153183370, 129156542039
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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For small n, this is equal to the binomial coefficient C(n,19). We have eleven possibilities for the first ball thrown in the first frame (gutter, 1, 2, ..., 9, strike) and the same possibilities occur for each of the other nine frames. So without even considering the second ball in each frame, at a minimum we have 11^10 possibilities. In fact, the true number of games is much larger due to the effect of the second ball in each frame. It's easy to show that the total number of possible games is: 66^9 * 241 = 5726805883325784576. The number of possible games with scores n = 0, 1, 2, 3, 4, 5, are: 1, 20, 210, 1540, 8855, 42504... The maximal number of possible games is for the score n = 77, namely 172542309343731946. Then there are a declining number of ways to get very high scores. For n = 288, 289, ..., 300 the totals are: 12, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1. The distribution of number of possible games as a function of n is not precisely symmetric about its maximal point. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 15 2005
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REFERENCES
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Cooper, C. N. and Kennedy, R. E. "A Generating Function for the Distribution of the Scores of All Possible Bowling Games." In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994.
Cooper, C. N. and Kennedy, R. E. "Is the Mean Bowling Score Awful?" In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994.
Curtis Cooper and Robert E. Kennedy, Mathematics Magazine, Vol. 63, No. 4, pp. 239-243. (Year?)
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LINKS
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Lee A. Newberg, Table of n, a(n) for n=0..300
Balmoral Software, All About Bowling Scores, 2005,
Eric Weisstein's World of Mathematics, Bowling.
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EXAMPLE
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The final terms are a(290) = 11, a(291) = ... = a(300) = 1.
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CROSSREFS
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Cf. A010972.
Sequence in context: A139620 A094311 A162640 this_sequence A010972 A126905 A022585
Adjacent sequences: A060850 A060851 A060852 this_sequence A060854 A060855 A060856
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KEYWORD
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nonn,fini,easy
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AUTHOR
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Micah Friese (friesem(AT)stolaf.edu), May 03 2001
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EXTENSIONS
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n = 77 peak value corrected by Lee A. Newberg (integer(AT)quantconsulting.com), Oct 30 2009
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