Difference between revisions of "2008 AMC 12B Problems/Problem 1"
(New page: ==Problem== A basketball player made <math>5</math> baskets during a game. Each basket was worth either <math>2</math> or <math>3</math> points. How many different numbers could represent ...) |
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<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math> | <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math> | ||
| − | ==Solution== | + | ==Solution 1== |
| − | If the basketball player makes <math>x</math> three-point shots and <math>5-x</math> two-point shots, he scores <math>3x+2(5-x)=10+x</math> points. Clearly every value of <math>x</math> yields a different number of total points. Since he can make any number of three-point shots between <math>0</math> and <math>5</math> inclusive, the number of different point totals is <math>6 \Rightarrow E</math>. | + | If the basketball player makes <math>x</math> three-point shots and <math>5-x</math> two-point shots, he scores <math>3x+2(5-x)=10+x</math> points. Clearly every value of <math>x</math> yields a different number of total points. Since he can make any number of three-point shots between <math>0</math> and <math>5</math> inclusive, the number of different point totals is <math>6 \Rightarrow E</math>. |
| + | ===Solution 2=== | ||
| + | Stars and bars can also be utilized to solve this problem. Since we need to decide what number of 2's and 3's are scored, and there are a total of 5 shots. It can be written like such: _ _ _ | _ _. Solving this, we get <math>6 \Rightarrow E</math>. | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2008|ab=B|before=First question|num-a=2}} | {{AMC12 box|year=2008|ab=B|before=First question|num-a=2}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 15:02, 19 August 2017
Contents
Problem
A basketball player made 
baskets during a game. Each basket was worth either 
or 
points. How many different numbers could represent the total points scored by the player?
Solution 1
If the basketball player makes 
three-point shots and 
two-point shots, he scores 
points. Clearly every value of yields a different number of total points. Since he can make any number of three-point shots between 
and inclusive, the number of different point totals is 
.
Solution 2
Stars and bars can also be utilized to solve this problem. Since we need to decide what number of 2's and 3's are scored, and there are a total of 5 shots. It can be written like such: _ _ _ | _ _. Solving this, we get .
See Also
| 2008 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by First question |
Followed by Problem 2 |
| 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 AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
