Difference between revisions of "2014 AMC 8 Problems/Problem 16"
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− | + | ==Problem== | |
+ | The "Middle School Eight" basketball conference has <math>8</math> teams. Every season, each team plays every other conference team twice (home and away), and each team also plays <math>4</math> games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams? | ||
− | + | <math>\textbf{(A) }60\qquad\textbf{(B) }88\qquad\textbf{(C) }96\qquad\textbf{(D) }144\qquad \textbf{(E) }160</math> | |
+ | |||
+ | ==Solution== | ||
+ | |||
+ | Within the conference, there are 8 teams, so there are <math>\dbinom{8}{2}=28</math> pairings of teams, and each pair must play two games, for a total of <math>28\cdot 2=56</math> games within the conference. | ||
+ | |||
+ | Each team also plays 4 games outside the conference, and there are 8 teams, so there are a total of <math>4\cdot 8 =32</math> games outside the conference. | ||
+ | |||
+ | Therefore, the total number of games is <math>56+32 = \boxed{88}</math>, so <math>\boxed{\text{(B)}}</math> is our answer. | ||
+ | |||
+ | ==Video Solution (CREATIVE THINKING)== | ||
+ | https://youtu.be/G2Akf39uwcE | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | |||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/Zhsb5lv6jCI?t=991 | ||
+ | |||
+ | https://youtu.be/w7Y-iq_kEaY ~savannahsolver | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC8 box|year=2014|num-b=15|num-a=17}} | ||
+ | {{MAA Notice}} |
Latest revision as of 11:19, 2 July 2023
Problem
The "Middle School Eight" basketball conference has teams. Every season, each team plays every other conference team twice (home and away), and each team also plays
games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?
Solution
Within the conference, there are 8 teams, so there are pairings of teams, and each pair must play two games, for a total of
games within the conference.
Each team also plays 4 games outside the conference, and there are 8 teams, so there are a total of games outside the conference.
Therefore, the total number of games is , so
is our answer.
Video Solution (CREATIVE THINKING)
~Education, the Study of Everything
Video Solution
https://youtu.be/Zhsb5lv6jCI?t=991
https://youtu.be/w7Y-iq_kEaY ~savannahsolver
See Also
2014 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.