how many questions to be answered

[veenu08]

In a certain test, a student must answer 8 out of 10 questions. How many choices are there if at least 4 of the first five questions must be answered?

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Hi veenu08,

If this is a GMAT question, then can you include the ENTIRE question, including answer choices? This question appears incomplete though. Is the student allowed to "skip" any questions. Is each question that is answered either "right" or "wrong"? Etc.

GMAT assassins aren't born, they're made,
Rich

[Matt@VeritasPrep]

Hi Veenu!

Let's simplify the question a bit first.

If I have to answer 8 out of 10 questions (with no restrictions), and the order in which I'm answering doesn't matter (and it doesn't seem to here), then the question is just a combinatorics question: how many ways can I choose 8 things out of 10?

But we DO have a restriction here, so let's explore the various scenarios. (I'm assuming here that the student will answer EXACTLY 8 questions, not 8, 9, or 10.)

SCENARIO 1:
I answer exactly 4 of the first 5 questions. In this case, I have to also answer exactly 4 of the OTHER 5 questions. So there are (5! / 4!) or 5 ways to pick 4 of the first 5, and (5! / 4!) or 5 ways to pick 4 of the remaining 5. That means I have 5 * 5 = 25 ways to answer my 8 questions.

SCENARIO 2:
I answer all 5 of the first 5 questions. In this case, I have to answer 3 of the remaining 5 questions, and there are (5! / (2! * 3!)) ways of doing that, or 10 ways. Since this is the only variable in this scenario, this is my answer: 10 ways.

These are my only two scenarios, so I have 25 + 10 or 35 total ways of answering exactly 8 questions, given the restriction that I must answer 4 of the first 5.