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.)

(In reply to Intuitive Representation IMHO by nikki)

Nikki,

My solution was using your

| | | | |O| |OOO|OOOOOO| | |

idea (except you should only have 9 dividers, not 11 as above).

Since it's hard to count how many of the above configurations there are, I counted instead the number of configurations with 20 balls with the restriction that there must be at least one divider between each pair of balls.  That number is easier to count.

I then argued that these two numbers should be the same.  Hopefully that makes some sense - let me know if my solution makes sense now. 

Posted by David Shin on 2004-09-23 22:51:37