In a true-false test of (6) questions, exactly (4) answers must be true and the first question must be false. How many answer patterns are possible?

Correct answer: (5)

The direct answer is option B, 5. A true-false answer pattern has two choices at every question: True or False. However, the first answer is already fixed as False, so it does not need to be chosen. Exactly four answers must be True. They must therefore be selected from questions 2, 3, 4, 5 and 6, which are five questions. Choose the one question that will be False, or equivalently choose four True answers: \binom{5}{4}=5. Thus there are 5 patterns. Option A, 10, would come from an incorrect selection count and does not respect the exact condition. Option B, 5, is correct because it counts the possible placements of four True answers among the remaining five positions. Option C, 15, overcounts the patterns, perhaps by using an unsuitable combination. Option D, 20, is also too large and ignores that the first position is fixed. Memory cue: fix the forced answer first, then count only the unfixed positions.