document.write( "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?
\n" ); document.write( "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. \n" ); document.write( "
Algebra.Com's Answer #27197 by kev82(151)\"\" \"About 
You can put this solution on YOUR website!
Hi,\r
\n" ); document.write( "\n" ); document.write( "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\r
\n" ); document.write( "\n" ); document.write( "0 then N(s,d) = N(s,d-1) (still need to get s but with one less dart)
\n" ); document.write( "1 then N(s,d) = N(s-1,d-1) (scored one, so need to score s-1 with one less dart)
\n" ); document.write( "2 then N(s,d) = N(s-2,d-1) (scored 2 so need s-2 with one less dart)
\n" ); document.write( "3 then N(s,d) = N(s-3,d-1) (...)\r
\n" ); document.write( "\n" ); document.write( "But we could score any of these, so\r
\n" ); document.write( "\n" ); document.write( "\r
\n" ); document.write( "\n" ); document.write( "Where N(0,0)=1, N(s<=0,d)=0, and N(s,d<=0)=0\r
\n" ); document.write( "\n" ); document.write( "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.\r
\n" ); document.write( "\n" ); document.write( "Here is the output of the program. The rows are the score and the colums are the number of darts.\r
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document.write( "        0       1       2       3       4       5       6\r\n" );
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document.write( "0|      1       1       1       1       1       1       1\r\n" );
document.write( "1|      0       1       2       3       4       5       6\r\n" );
document.write( "2|      0       1       3       6       10      15      21\r\n" );
document.write( "3|      0       1       4       10      20      35      56\r\n" );
document.write( "4|      0       0       3       12      31      65      120\r\n" );
document.write( "5|      0       0       2       12      40      101     216\r\n" );
document.write( "6|      0       0       1       10      44      135     336\r\n" );
document.write( "7|      0       0       0       6       40      155     456\r\n" );
document.write( "8|      0       0       0       3       31      155     546\r\n" );
document.write( "9|      0       0       0       1       20      135     580\r\n" );
document.write( "10|     0       0       0       0       10      101     546\r\n" );
document.write( "11|     0       0       0       0       4       65      456\r\n" );
document.write( "12|     0       0       0       0       1       35      336\r\n" );
document.write( "13|     0       0       0       0       0       15      216\r\n" );
document.write( "14|     0       0       0       0       0       5       120\r\n" );
document.write( "15|     0       0       0       0       0       1       56\r\n" );
document.write( "16|     0       0       0       0       0       0       21\r\n" );
document.write( "17|     0       0       0       0       0       0       6\r\n" );
document.write( "18|     0       0       0       0       0       0       1\r\n" );
document.write( "19|     0       0       0       0       0       0       0\r\n" );
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\r
\n" ); document.write( "\n" ); document.write( "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.\r
\n" ); document.write( "\n" ); document.write( "Hope that helps,
\n" ); document.write( "Kev \n" ); document.write( "
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