A test has 200 questions. Each question has 5… | GMAT Sprint

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PS · Problem Solving

705-805

Combinations

A test has 200 questions. Each question has 5 options, but only 1 option is correct. If test-takers mark the correct option, they are awarded 1 point. However, if an answer is incorrectly marked, the test-taker loses 0.25 points. No points are awarded or deducted if a question is not attempted. A certain group of test-takers attempted different numbers of questions, but each test-taker still received the same net score of 40. What is the maximum possible number of such test-takers?

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Explanation

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Official Explanation

Official

Let

x

= number of correct answers,

y

= number of incorrect answers.

Net score =

x - 0.25y = 40

Multiply by 4:

4x - y = 160

Total attempted =

x + y = n

From

4x - y = 160

and

x + y = n

, add equations:

5x = 160 + n

x = \frac{160 + n}{5}

Since

x

must be an integer,

160 + n

must be divisible by 5 →

n

must be a multiple of 5.

Also

x \leq n \leq 200

and

x \geq 0

y \geq 0

.

From

x = \frac{160 + n}{5}

and

y = n - x = n - \frac{160 + n}{5} = \frac{4n - 160}{5}

, we need

y \geq 0

→

4n - 160 \geq 0

→

n \geq 40

.…

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What this question tests

What it tests

Whether you can count arrangements and selections correctly — permutations vs. combinations.

Common trap

Overcounting: treating a selection as ordered when order does not matter, or forgetting to divide by duplicates.