The number of permutations of the letters of the word LUMINARY such that niether LURY nor MINA occur together?
a. 46800 b. 24680 c. 40086 d none of these

• If we observe there are 8 letters in the given word LUMINARY.
  Now we will find the number of permutations in which either LURY or MINA occurs.
  L U R Y _ _ _ _ we will fix LURY on 4 places out of 8 places. so remaining letters can be arranged on remaining places in 4! ways. And LURY can be kept on 5 places.
  hence the total number of ways in which we can get LURY is 5 x 4! ways.
  
  Similar logic for MINA.
  Total number of ways in which we can get MINA is 5 x 4! ways.
  
  Number of permutations in which LURY and MINA doesnt occur = Total number of permutations - Number of permutations i which LURY and MINA occurs
  
  => Number of ways in which LURY and MINA doesnt occur = 8! - (5 x 4! + 5 x 4!) = 40080 ways
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