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Sally constructed a dart board. The possible score 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 throwing 6 darts at the board. The scores from the 6 throws are added. In how many ways can a total score of 15 or 16 be obtained?
NOTE:The totol score obtained from 1+0+0+3+3+2 is considered different from the total score obtained from 0+0+1+3+3+2
Can get 15 by getting
3,3,3,3,3,0 giving 6 possible arrangements
3,3,3,3,2,1 giving 5 + 5 + 5 + 5 + 5 + 5 = 30 possible arrangemnts
3,3,3,2,2,2 giving 4 + 3 + 2 + 1 + 3 + 2 + 1 + 2 + 1 + 1 = 20 possible arrangements
Giving 56 possibles for score of 15
Can get 16 by getting
3,3,3,3,3,1 giving 6 possible arrangements
3,3,3,3,2,2 giving 5 + 4 + 3 + 2 + 1 = 15 possible arrangements
Giving 21 possibles for score of 16
Gives 77 possibles for score of 15 and 16
The trick is to think systematically (though I believe there are formulas out there that will provide short cuts for working out each possible arrangements).
Fix one number and work out the posssible arrangements. Then move it to another position and fix it and work out possibles, etc, etc