A paper had ten questions. Each could only be answered as True (T) or False (F). Each candidate answered all the questions. Yet, no two candidates wrote the answers in an identical sequence. How many different sequences of answers are possible?

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Q: 28 (IAS/2010)
A paper had ten questions. Each could only be answered as True (T) or False (F). Each candidate answered all the questions. Yet, no two candidates wrote the answers in an identical sequence. How many different sequences of answers are possible?

question_subject: 

Maths

question_exam: 

IAS

stats: 

0,9,13,4,4,5,9

keywords: 

{'many different sequences': [0, 0, 1, 0], 'identical sequence': [0, 0, 1, 0], 'candidates': [3, 4, 1, 6], 'candidate': [1, 0, 0, 0], 'questions': [2, 0, 5, 3], 'paper': [1, 0, 4, 7], 'answers': [0, 0, 1, 0]}

Each question on the paper can be answered in two ways: True (T) or False (F). Since there are ten questions, to find out the number of different sequences of answers, we use the formula for calculating the number of outcomes in a situation with two possibilities (T or F), raised to the power of the number of instances (10 questions). This equates mathematically to 2^10.

Option 1 (20) and Option 2 (40) are not the correct answers because they are less than the total possible combinations of 2^10.

Option 3 (512) is the value of 2^9, not 2^10. So, it is close but not the correct number of sequences for 10 questions.

Option 4 (1024) is the correct answer because it is the value of 2^10, representing the number of different sequences for ten questions with two options each.