In a test consisting of 15 questions, 3 marks are awarded for a correc

Question

In a test consisting of 15 questions, 3 marks are awarded for a correct answer, 1 mark is deducted for an incorrect answer and no mark is awarded for a non-attempted question. If a student attempts at least one question in the paper, what is the number of distinct scores that he can get?

(A) 57
(B) 58
(C) 59
(D) 60
(E) 61

Are You Up For the Challenge: 700 Level Questions

(A) 57
(B) 58
(C) 59
(D) 60
(E) 61

Are You Up For the Challenge: 700 Level Questions

chetan2u · Accepted Answer

The least the student can get is -15 when he answers all wrong. 
The next possibility is 14 wrong, so -14.
Thus he can get each of the possible scores in the lower range.

What about upper end?
He can get max of 15*3 or 45 marks. 
But he cannot get 43 and 44 as he does not have any more questions for negative marking. 
He can get 42 marks by answering 14 right. He is left with one question, it can be left or answered incorrectly, so 42-1 or 41. 
But again he cannot score 40 as he does not have any more questions left to answer. 
Lesser than that he can get all possible values as his score. For example 13 correct and two not attempted or 13 correct and one not attempted and one wrong or 13 correct and two wrong.

Thus he can get all scores from -15 to 45 except 44, 43 and 40. => (45-(-15)+1)-3=61-3=58

B

desertEagle · Answer

Maximum Score obtainable = 15x3=45
Minimum Score Obtainable = 15x-1=-15

44, 43, 40 are not possible

So, scores possible = {-15,-14,-13,----45} - {40,43,44}
Nos. of score possible = 61-3 = 58

Csanamika · Answer

How do we know that only 44, 43, and 40 are not possible? When we solve it case by case for different numbers of correct, incorrect, and unanswered questions, we can arrive at this conclusion. But what is the shortest approach? What is the underlying thought behind it?

Kinshook · Answer

In a test consisting of 15 questions, 3 marks are awarded for a correct answer, 1 mark is deducted for an incorrect answer and no mark is awarded for a non-attempted question.

If a student attempts at least one question in the paper, what is the number of distinct scores that he can get?

Questions attempted = n; Correct = c ; Incorrect = n - c; Score = 3c - (n-c) = 4c - n 
Scores = {3n, 3n - 4, 3n - 8,.... , -n}

Impossible scores = {44,43,40} : 3 scores

All Scores = number of all scores from -15 to 45 - number of Impossible scores = 61 - 3 = 58

IMO B

egmat · Answer

Hi Csanamika,

You've verified it case by case; what you're missing is the one clean idea that makes  44 ,  43 , and  40  fall out instantly. It's already hiding inside chetan2u's and desertEagle's posts - let me make it explicit.

Start from the top and ask "how do I step down?"

The maximum is  45  (all  15  correct). Now ask: from a fully-correct paper, what are the  only  ways to lower your score?

- Turn one  correct  answer into an  unattempted  one - you lose the  +3 , so the score drops by  3 .
- Turn one  correct  answer into a  wrong  one - you lose the +3  and  pick up a  -1 , so the score drops by  4 .

There is no move that lowers your score by  1  or by  2 . So  every  score below the max must be:

45 - (some combination of 3's and 4's).

Which drops are impossible?

Now it's a tiny sub-question: which whole numbers  can't  be written as a sum of 3's and 4's?

-  1  - impossible - so  44  is impossible.
-  2  - impossible - so  43  is impossible.
-  5  - impossible (3+? and 4+? both overshoot) - so  40  is impossible.
-  3, 4, 6, 7, 8, ...  - all possible (6=3+3, 7=3+4, 8=4+4, and everything after keeps working) - so  42 ,  41 ,  39 ,  38 , ... are all reachable.

That's the whole thing: the only unreachable amounts are  1, 2, and 5 , which knock out exactly  44, 43, and 40 . Everything from  -15  up to  45  except those three is fair game -  61 - 3 = 58 .

Lock the idea in

Try the same rules with just  5  questions (max =  15 ). Below  15  you can again only subtract 3's and 4's, so you can't get 15-1 =  14 , 15-2 =  13 , or 15-5 =  10  - the same three gaps at the top. Same principle, smaller numbers.

Answer: B

In a test consisting of 15 questions, 3 marks are awarded for a correc

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