Difference between revisions of "2014 AMC 8 Problems/Problem 16"
Mathonator (talk | contribs) |
Pi is 3.14 (talk | contribs) |
||
| (13 intermediate revisions by 9 users not shown) | |||
| Line 3: | Line 3: | ||
<math>\textbf{(A) }60\qquad\textbf{(B) }88\qquad\textbf{(C) }96\qquad\textbf{(D) }144\qquad \textbf{(E) }160</math> | <math>\textbf{(A) }60\qquad\textbf{(B) }88\qquad\textbf{(C) }96\qquad\textbf{(D) }144\qquad \textbf{(E) }160</math> | ||
| + | |||
==Solution== | ==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 1 by OmegaLearn== | ||
| + | https://youtu.be/Zhsb5lv6jCI?t=991 | ||
| + | |||
| + | ==Video Solution (CREATIVE THINKING)== | ||
| + | https://youtu.be/G2Akf39uwcE | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | |||
| + | ==Video Solution 3== | ||
| + | https://youtu.be/Zhsb5lv6jCI?t=991 | ||
| + | |||
| + | https://youtu.be/w7Y-iq_kEaY ~savannahsolver | ||
| + | |||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2014|num-b=15|num-a=17}} | {{AMC8 box|year=2014|num-b=15|num-a=17}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 15:25, 29 September 2024
Contents
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 1 by OmegaLearn
https://youtu.be/Zhsb5lv6jCI?t=991
Video Solution (CREATIVE THINKING)
~Education, the Study of Everything
Video Solution 3
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. 
