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In how many ways can a test paper of six questions be attempted, each question being of a TRUE/FALSE type?

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if each question has three options (T, F, or Blank) then each additional question gives you three times the options you had before. in other words, `N(attempts) = 3×3×3×3×3×3 = 3⁶ = 729`
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If you attempt 1 question, there are 2 ways to answer that question but there are 6 questions that could have been chosen to answer. So for line 2, it should be 1\*2!\*(6C1). This is the error in the way you constructed your answer.

As another poster has pointed out, a simpler way to get to the answer is to consider that for every question, there are three options - don't answer, answer T or answer F

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