SOLUTION: Sally constructed a dart board. The possible scores are 0, 1, 2 or 3. A score of 0 is obtained if the dart misses the board. Sally challenges John to a game consisting of each pla

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Question 42057: Sally constructed a dart board. The possible scores are 0, 1, 2 or 3. A score of 0 is obtained if the dart misses the board. Sally challenges John to a game consisting of each player throwing6 darts at the board. The scores from 6 throws are added. In how many ways can a score of 15 or 16 can be obtained?
NOTE: The total score obtained from 1+0+0+3+3+2 is considered different from the total score obtained from 0+0+1+3+3+2.

Answer by kev82(151) About Me  (Show Source):
You can put this solution on YOUR website!
Hi,
Let N(s,d) be the number of ways to achieve a score of s using exactly d darts. In each throw we can get a score of either 0, 1, 2, or 3 so if we scored
0 then N(s,d) = N(s,d-1) (still need to get s but with one less dart)
1 then N(s,d) = N(s-1,d-1) (scored one, so need to score s-1 with one less dart)
2 then N(s,d) = N(s-2,d-1) (scored 2 so need s-2 with one less dart)
3 then N(s,d) = N(s-3,d-1) (...)
But we could score any of these, so

Where N(0,0)=1, N(s<=0,d)=0, and N(s,d<=0)=0
I'm sure that there is an exact formula for N, probably with lots of factorials, but in much less than the time it would take me to figure it out, I could write a computer program to do it for me. So I'll do that.
Here is the output of the program. The rows are the score and the colums are the number of darts.
        0       1       2       3       4       5       6

0|      1       1       1       1       1       1       1
1|      0       1       2       3       4       5       6
2|      0       1       3       6       10      15      21
3|      0       1       4       10      20      35      56
4|      0       0       3       12      31      65      120
5|      0       0       2       12      40      101     216
6|      0       0       1       10      44      135     336
7|      0       0       0       6       40      155     456
8|      0       0       0       3       31      155     546
9|      0       0       0       1       20      135     580
10|     0       0       0       0       10      101     546
11|     0       0       0       0       4       65      456
12|     0       0       0       0       1       35      336
13|     0       0       0       0       0       15      216
14|     0       0       0       0       0       5       120
15|     0       0       0       0       0       1       56
16|     0       0       0       0       0       0       21
17|     0       0       0       0       0       0       6
18|     0       0       0       0       0       0       1
19|     0       0       0       0       0       0       0

Using this we can see that the number of ways to score 15 with 6 dart is 56, and the number of ways to score 16 is 21. So 56+21=77 which is your answer.
Hope that helps,
Kev