JEE Advanced · 2020 · 27 Sep 2020 · Paper 2 · Permutations and Combinations

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An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021 is ____

Answer: 495

Solution

We need to select 4 days out of 15 such that no two are consecutive. This is equivalent to choosing 4 items from

15 - 4 + 1 = 12

positions (using the stars-and-bars method for non-consecutive selections). The number of ways is

^{12} C_{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = \frac{11880}{24} = 495

JEE Advanced · 2020 · 27 Sep 2020 · Paper 2 · Numerical answer

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