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📐 Mathematics  ·  Permutations and Combinations  ·  JEE

The number of positive integers less than 1000 divisible by neither 5 nor 7 is:

Answer: 686.

  • A 640
  • B 686
  • C 714
  • D 648

Correct answer: B. 686

Explanation: Among 1–999: divisible by 5 is 199, by 7 is 142, by 35 is 28. By 5 or 7 = 313, so neither = 999 − 313 = 686.

StartABCBCACAB6 ordered outcomes (permutations); pairing AB/BA etc gives 3 combinations

Counting tree for selecting 2 items from {A, B, C} without repetition: 3 x 2 = 6 ordered arrangements.

Concept context

Counting, arrangements, and selections

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