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 _____.

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Solution

To select 4 days out of 15 such that no two visits take place on consecutive days= $^{15} C_{4}$
Ref. image

$\frac{15 - 4}{G a p s} \Rightarrow \frac{^{11 + 1} C_{4}}{s e l e c t i o n} \Rightarrow^{12} C_{4}$ = $495$

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To select 4 days out of 15 such that no two visits take place on consecutive days=15C4Ref. image 15−4Gaps⇒11+1C4selection⇒12C4=495

To select 4 days out of 15 such that no two visits take place on consecutive days= $^{15} C_{4}$
Ref. image

$\frac{15 - 4}{G a p s} \Rightarrow \frac{^{11 + 1} C_{4}}{s e l e c t i o n} \Rightarrow^{12} C_{4}$ = $495$