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Tough question?
Sally constucted a dart board. the possible socres are 0, 1, 2 or 3. A score of zero 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 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. I think I need a formular, but what is it?
All combinations of:
3,3,3,3,3,1 --- 6nPr6/(5nPr5)(1nPr1) = 6,
where 6nPr6 = total # of rearrangements, 5nPr5 = possible repeated rearrangements of the 3's, 1nPr1 = the possible repeated rearrangement of the 1 (it equals 1 because one # can't be rearranged with itself, so it's a pointless calculation). Divide the total arrangements by all the possible repeat rearrangements to find the total number of non-repeat rearrangements for this combination.
3,3,3,3,3,0 --- (6nPr6)/(5nPr5) = 6
3,3,3,3,2,2 --- (6nPr6)/(4nPr4)(2nPr2) = 15
3,3,3,3,2,1 --- (6nPr6)/(4nPr4) = 30
3,3,3,2,2,2 --- (6nPr6)/(3nPr3)(3nPr3) = 20
No other combinations of 6 numbers will give you a total of 15 or 16
Find the total number of non-repeat rearrangements for each set (which I've done above). And add up the results: 6 + 6 + 15 + 30 + 20 = 77
77
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