Question

If a person throws 3 darts at a dartboard, how many possible combinations are there to get a sum of 34?

Answer 1

A dart board has sectors numebered from 1 to 20.
We require combinations, so order does not count.

Here's how we could proceed:
1. to avoid repetitions, we <i>assume</i> the smallest number is thrown first, then the next smallest, and the largest numeber.
2. There are no limit to repetitions, e.g. (7,7,20) is a valid combination.
3. Make a table of possible triplets
(1,13,20)...(7,7,20) => 7 triplets
4. then reduce the third number by one and repeat.
(1,14,19)...(7,8,19) => 7 triplets
(1,15,18)...(8,8,18) => 8 triplets
5. continue working until we get (10,12,12)...(11,11,12) => 2 triplets
6. Sum the count of all triplets, which is the correct answer.

Hint:to help you check your answer, it should be below 100 and be divisible by 9.