Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 21"
(→Solution) |
|||
| (3 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
| + | Suppose that each pair of eight tennis players either played exactly one game last week or did not play at all. Each player participated in all but 12 games. How many games were played among the eight players? | ||
| − | <center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ } </math></center> | + | <center><math> \mathrm{(A) \ }10 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ }14 \qquad \mathrm{(D) \ }16 \qquad \mathrm{(E) \ }18 </math></center> |
== Solution == | == Solution == | ||
| + | Since each player played in all but 12 games, each player played in the same number of games, say <math>g</math>. Since each game was played by 2 players, this means there were <math>\frac12\cdot 8g = 4g</math> games played. However, by the given there were also <math>g + 12</math> games played. Thus <math>4g = g + 12</math>, <math>g = 4</math> and the answer is <math>16 \Longrightarrow \mathrm{(D)}</math>. | ||
| − | + | ---- | |
| − | * [[University of South Carolina High School Math Contest/1993 Exam]] | + | |
| + | * [[University of South Carolina High School Math Contest/1993 Exam/Problem 20|Previous Problem]] | ||
| + | * [[University of South Carolina High School Math Contest/1993 Exam/Problem 22|Next Problem]] | ||
| + | * [[University of South Carolina High School Math Contest/1993 Exam|Back to Exam]] | ||
| + | |||
| + | [[Category:Introductory Combinatorics Problems]] | ||
Latest revision as of 16:38, 18 August 2006
Problem
Suppose that each pair of eight tennis players either played exactly one game last week or did not play at all. Each player participated in all but 12 games. How many games were played among the eight players?
Solution
Since each player played in all but 12 games, each player played in the same number of games, say 
. Since each game was played by 2 players, this means there were 
games played. However, by the given there were also 
games played. Thus 
, 
and the answer is 
.
