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A test has 10 True/False questions. The number of ways a student can answer such that at least one answer is wrong equals which of the following? (A) The number of sequences when a fair coin is tossed 10 times and both heads and tails appear at least once. (B) The number of ways to answer a question with 10 options where one or more options can be correct. (C) The number of ways to choose a sum of money using 10 coins of different denominations taking some or all at a time. (D) The number of different selections of 10 identical objects taking some or all.
- (A) Number of coin-toss sequences of length 10 containing both H and T.
- (B) Number of ways to answer a 10-option MCQ with one or more correct answers.
- (C) Number of ways to select some or all of 10 coins of different denominations.
- (D) Number of selections of 10 identical objects taking some or all at a time.
Correct answer: (C) Number of ways to select some or all of 10 coins of different denominations.
Solution
With 10 T/F questions, total answer sequences = 2¹⁰ = 1024. Sequences with at least one wrong answer = 1024 - 1 = 1023. For option C: each coin is either selected or not, giving 2¹⁰ subsets, minus the empty set = 2¹⁰ - 1 = 1023. This matches.
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