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

Question

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 ______. [2020]

Smart Achievers · Accepted Answer

(495)

We know that total number of ways of selection of $r$ days out of $n$ days such that no two of them are consecutive $= {}^{n - r + 1}C_{r}$

$∴$ Selection of 4 days out of 15 days such that no two of them are consecutive $= {}^{15 - 4 + 1}C_{4} = {}^{12}C_{4}$

$= \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2} = 11 \times 5 \times 9 = 495$

Solution

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Solution

Ans. (495) We know that total number of ways of selection of r days out of n days such that no two of them are consecutive =Cr n-r+1 ∴ Selection of 4 days out of 15 days such that no two of them are consecutive =C415-4+1=C412 =12×11×10×94×3×2=11×5×9=495

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