A test had 12 problems, and each problem is with five marks. Full marks are given for a correct answer, two marks given if there is no answer, and no marks are given for an incorrect answer. Some scores between 1 and 60 or impossible to get on this test. What is the sum of these impossible to get scores?
If there are x correct answers and y unanswered, then the score is 5x+2y
So just list all the combinations where x+y=12 to get all the possible scores.
I think there is only one combination which is (5x2)+(2x1)=12. I don't know what to do from there.
Actually, I oversimplified. We need x+y <= 12
So, there are 13*14/2 = 91 combinations in all
x y score
0 0 0
0 1 2
...
0 12 24
1 0 5
...
1 11 27
2 0 10
2 1 12
...
2 10 30
...
This could be done in one line using the APL language, but it gets complicated just doing some sums because there will be several combinations that produce the same score.
I'd write a small program to do it, but hey -- not my job. I don't see an easy number-theoretic way to solve this. Just brute force.
To find the sum of the impossible scores, we need to determine which scores are possible on this test and remove them from the total sum.
Let's break down the possibilities for each problem:
- A correct answer gives 5 marks
- No answer gives 2 marks
- An incorrect answer gives 0 marks
Since there are 12 problems, the total marks for a perfect score are 12 x 5 = 60.
To find the impossible scores, we need to consider the values that cannot be obtained. Here are some observations to guide us:
- The lowest possible score is 0 (if all answers are incorrect).
- The highest possible score is 60 (if all answers are correct).
- The scores between 0 and 60 can be categorized into the following ranges:
- 0 to 5: Can be obtained by answering a single problem incorrectly while leaving the rest blank (making sure it is the only question answered incorrectly).
- 7 to 12: Can be obtained by answering a single problem correctly while leaving the rest blank (making sure it is the only question answered correctly).
- 14 to 17: Can be obtained by answering two problems correctly while leaving the rest blank (making sure they are the only questions answered correctly).
- 19 to 24: Can be obtained by answering two problems incorrectly while leaving the rest blank (making sure they are the only questions answered incorrectly).
- The pattern continues in increments of 2 until the upper limit of 60.
To calculate the sum of impossible scores, we need to find the range of possible scores and subtract it from the total possible score (60).
Using the observations mentioned above, we can calculate the sum of the impossible scores:
1. Calculate the number of possible scores: The range of possible scores consists of every possible sum in the specified ranges. The count can be obtained by calculating the difference between the highest and lowest possible scores and then adding 1 (to include both endpoints).
The count = highest_possible_score - lowest_possible_score + 1
In our case:
highest_possible_score = 60
lowest_possible_score = 0
count = 60 - 0 + 1 = 61
2. Calculate the sum of the possible scores:
sum_of_possible_scores = (count / 2) * (lowest_possible_score + highest_possible_score)
In our case:
sum_of_possible_scores = (61 / 2) * (0 + 60) = 1830
3. Calculate the sum of the impossible scores:
sum_of_impossible_scores = total_possible_score - sum_of_possible_scores
In our case:
sum_of_impossible_scores = 60 - 1830 = -1770
Since we cannot have negative scores, the sum of all impossible scores is 0.
Therefore, the sum of the impossible scores on this test is 0.