Problem 27 · 2011 Math Kangaroo
Stretch
Counting & Probability
careful-counting
An archer tries his skill on the target shown on the right. With each of his three arrows he always hits the target. How many different total scores can he make with three arrows?
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Answer: C — 19
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Hint 1 of 2
Each arrow scores 1, 3, 7 or 12, and the three scores add up; repeats are allowed.
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Hint 2 of 2
List the possible totals for all triples and count the distinct values.
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Approach: enumerate distinct sums of three values from {1,3,7,12}
- Any three arrows give a total a+b+c with each of a,b,c in {1, 3, 7, 12}.
- Listing every combination yields totals 3,5,7,9,11,13,14,15,16,17,18,20,21,22,25,26,27,31,36.
- That is 19 different scores.
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