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?
Just list them:
3+3+3+3+3
3+3+3+3+2+1
...
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1
Is it 15 WAYS
To find the number of ways Joe can score 15 points, we can break it down into three cases based on the number of each type of shot:
Case 1: All long shots (3 points)
In this case, Joe would need to make 5 long shots to score 15 points. Since each long shot is worth 3 points, there is only 1 way to score 15 points using all long shots.
Case 2: Combination of long shots (3 points) and field goals (2 points)
Since Joe has already made 15 points, he cannot make any more long shots (as they are worth 3 points each) because it would give him more than 15 points. Therefore, we are left with only 0, 2, 4, 6, 8, 10, 12, or 14 points from long shots. For each of these possibilities, we can calculate the remaining points needed from field goals.
- For 0 points from long shots: Joe needs to score 15 points in field goals. The number of ways to combine field goals to make 15 points can be calculated using a counting technique called partitions. However, the calculation is quite intricate for larger values, so we can use a dynamic programming approach instead. The number of ways to make 15 points in field goals is 5.
- For 2 points from long shots: Joe needs to score 13 points in field goals. Similarly, the number of ways to make 13 points in field goals is 5.
- And so on...
Using this logic, let's calculate the number of ways for each possibility:
- For 0 points from long shots: 1 way (5 field goals)
- For 2 points from long shots: 1 way (4 field goals, 1 long shot)
- For 4 points from long shots: 2 ways (3 field goals, 2 long shots) or (1 field goal, 3 long shots)
- For 6 points from long shots: 3 ways (2 field goals, 3 long shots), (4 field goals, 1 long shot), or (6 field goals)
- For 8 points from long shots: 4 ways (1 field goal, 5 long shots), (3 field goals, 4 long shots), (5 field goals, 2 long shots), or (7 field goals, 1 long shot)
- For 10 points from long shots: 5 ways (0 field goals, 5 long shots), (2 field goals, 4 long shots), (4 field goals, 3 long shots), (6 field goals, 1 long shot), or (8 field goals)
- For 12 points from long shots: 6 ways (1 field goal, 3 long shots), (3 field goals, 2 long shots), (5 field goals, 1 long shot), (7 field goals), (9 field goals, 1 long shot), or (11 field goals)
- For 14 points from long shots: 7 ways (0 field goals, 4 long shots), (2 field goals, 3 long shots), (4 field goals, 2 long shots), (6 field goals, 1 long shot), (8 field goals), (10 field goals, 1 long shot), or (12 field goals)
Adding up the number of ways from each case:
Case 1: 1 way (all long shots) = 1 way
Case 2: 1 way (0 points from long shots) + 1 way (2 points from long shots) + 2 ways (4 points from long shots) + 3 ways (6 points from long shots) + 4 ways (8 points from long shots) + 5 ways (10 points from long shots) + 6 ways (12 points from long shots) + 7 ways (14 points from long shots) = 29 ways
Therefore, there is a total of 1 + 29 = 30 ways for Joe to score 15 points.
To find the number of ways Joe can score 15 points, we can use a technique called a "brute-force" approach or an iterative method.
We can start by considering all possible combinations of Joe's shots to reach a total of 15 points, taking into account that there are three different types of shots: long shots, field goals, and free throws.
Let's break it down step by step, starting with the possible number of long shots:
1. Joe can make 0, 1, 2, 3, 4, or 5 long shots. Since each long shot is worth 3 points, we need to find the remaining points Joe needs to score in each case.
a. If Joe makes 0 long shots, he will have to score 15 points with field goals and free throws.
b. If Joe makes 1 long shot, he will have to score 12 points with field goals and free throws.
c. If Joe makes 2 long shots, he will have to score 9 points with field goals and free throws.
d. If Joe makes 3 long shots, he will have to score 6 points with field goals and free throws.
e. If Joe makes 4 long shots, he will have to score 3 points with field goals and free throws.
f. If Joe makes 5 long shots, he won't need any field goals or free throws.
2. For each case above, we can calculate the possible number of field goals using a similar approach:
a. If Joe has 0 long shots and 0 field goals, he will need to score 15 points with free throws.
b. If Joe has 0 long shots and 1 field goal, he will need to score 13 points with free throws.
c. If Joe has 0 long shots and 2 field goals, he will need to score 11 points with free throws.
d. Continuing this process, we can calculate the remaining cases.
3. Finally, we consider the number of free throws needed to reach the remaining points.
By continuing this step-by-step calculation, we can count the number of ways Joe can score 15 points.