19. Three boys played a "Word" game in which they each wrote down ten words. For each word a boy wrote, he scored three points if neither of the other boys had the same word; he scored one point if only one of the other boys had the same word. No points were awarded for words which all three boys had. When they added up their scores, they found that they each had different scores. Sam had the smallest score (19 points), and James scored the most. How many points did James score?
A 20
B 21
C 23
D 24
E 25
SolUtion
E

Let $s$ be the number of times that Sam scored 3 points. If $s \leq 4$ then his maximum score would be $3 \times 4+6=18$ which is too small. If $s \geq 7$ then his minimum score would be $3 \times 7=21$ which is too big. Hence $s$ is either 5 or 6 .

Suppose $s=6$. So Sam had 6 words not shared with the other boys, giving him 18 points. To get a score of 19 , he must have shared one word with one other boy and picked three words shared with them both. That means that the other two boys also scored 0 points for those three words. To score more than 19 they must have scored 3 points for each of their 7 remaining words. However one of them has shared a word with Sam so scored a maximum of 19 points. Hence $s \neq 6$.

Hence $s=5$, giving Sam 15 points for these 5 words. To score 19, Sam must have scored 1 point for another four words and there must have been one word shared by all three boys. Each of the other two must have scored 3 points at least 6 times in order to score more than Sam. Also each scored 0 points for one word. That determines the scores for seven of each boy's words. Between the two, they shared a word with Sam 4 times; and these could be divided between them either 2 each or 1 and 3. The former case leaves only one word each to fix and this could be either 1 point each or 3 points each. But that would give the two boys the same score. Therefore one boy had 3 words shared with Sam and so a score of 21 . The other had only one word shared with Sam and the remaining two words must have scored 3 points, giving a total of 25 . So James scored 25 points.