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.)
This is the classic problem of putting 9 dividers into a row of 10 balls. The 10 bins formed will then correspond to the 10 point values scored on each shot.
This is equivalent to putting 9 dividers into a row of 20 balls with the condition that every bin has to have at least one ball. To see this, note that removing one ball from each bin of this new configuration yields a configuration satisfying the original constraint.
We can then compute the desired answer by noting that there are 19 possible places to put the 9 dividers. This gives 19 choose 9, which you can type directly into Google to get the answer of 92,378.
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