Joe made 15 points in a basketball game, 3 points are given for a long shot, 2 points given for a field goal, and 1 point is given for a free throw. In how many ways can Joe score 15 points?
My answer: 6 ways
is this correct or incorrect please help
1. 5 Long shots.
2. 4 Long shots, 1 field goal, 1 free throw.
3. 4 Long shots and 3 free throw.
4. 3 Long shots, 3 field goals.
5. 3 Long shots, 2 field goals, and 2 free throws.
6. 3 Long shots, 1 field goal, 4 free throws.
7. 3 Long shots, 6 free throws.
8. 2 long shots, 4 field goals, 1 free throw.
9. 2 Long shots, 3 field goals, 3 free throws.
10. 2 Long shots, 2 field goals, and 5 free throws.
11. 2 Long shots, 1 field goal, and 7 free throws.
12. 2 Long shots, 9 free throws.
13. 1 Long shot, 4 field goals, and 4 free throws.
14. 7 field goals, 1 free throw.
15. 6 field goals, 1 field goal, 1 free throws.
16. 5 field goals, 5 free throws.
17. 4 field goals, 7 free throws.
18. 3 field goals, 9 free throws.
19. 2 field goals, 11 free throws.
20. 1 field goal, 13 free throws.
21. 15 free throws.
21 Ways to score 15 points.
To determine the number of ways Joe can score 15 points, we can use a method called "stars and bars." In this approach, we imagine the points scored as stars and separate them into categories using bars.
Since Joe can score three different types of shots (long shots, field goals, and free throws), we can represent the points as follows:
lll | l | fffffff | fffffff | fffffff
In this representation, "lll" represents three long shots (9 points in total), "l" represents one long shot (3 points), and "fffffff" represents three field goals (6 points) per occurrence. We have divided them using bars to separate the shots.
Now, we can observe that the number of ways Joe can score 15 points is equivalent to finding the number of ways these bars and stars can be arranged. In this case, we have 5 elements (3 bars and 2 stars), and we need to arrange these elements.
This can be done using the formula for combinations with repetition, which is given by:
C(n + k - 1, k) = C(5, 2) = 10
Therefore, there are 10 distinct ways Joe can score 15 points, not 6.
Therefore, your answer of 6 ways is incorrect. The correct answer is 10 ways.