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A student writes $40$ chemistry True/False questions in an exam. Among these questions, $17$ are true. If the questions can be positioned in any order, then how many different answer keys are possible?

  1. $40! ^* (23!^* 17! )$
  2. $40^* (23!^*17!)$
  3. $40!/(23!^*17!)$
  4. $40!^* C(40,17)$
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question is to count number of answer keys possible , there are 40 questions, now there will be 40 true and false in total in any answer key,
out of 40,it is given that 17 are true and then 23 are false, So answer to the question should be arrangements of 40 objects where 23 are of similar one type and 17 are of similar another type,
so answer is 40!/(23! * 17!)
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