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| More Archery Practice (Posted on 2004-09-21) |
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Someone shot 10 arrows at a target with 10 concentric rings, each being worth a different integer number of points from 1 to 10. How many different ways are there of scoring 10 points by doing this? (Note that not all the arrows have to hit the target and that order matters; 6 first then 4 is different from 4 first then 6. Also note that two or more arrows may hit the same ring.)
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Submitted by Gamer
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Rating: 3.0000 (5 votes)
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Solution:
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The way I found to do this is quite cool. Imagine you have 1s and barriers. Each barrier can be off or on.
1|1|1|1|1|1|1|1|1|1 would represent 10 1 shots.
1|1 1 1|1 1 1|1 1|1 would represent 1,3,3,2,1.
1|1 1 1 1 1 1 1|1 1 would represent 1,7,2.
By using this, it is evident that there are 512 different ways to shoot, or 2 to the 9th power.
If you count arrows that don't hit the target as 0, then it will be like putting 9 barriers into 10 targets such that all the targets are used, and the number of ways for this is 19C9 or 92,378. A more complete explanation is here.
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