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Find the value of given expression.

\(\left( {1 + \;\frac{1}{P}} \right)\left( {1 + \;\frac{1}{{P + 1}}} \right)\left( {1 + \;\frac{1}{{P + 2}}} \right)\left( {1 + \;\frac{1}{{P + 3}}} \right)\left( {1 + \;\frac{1}{{P + 4}}} \right)\left( {1 + \;\frac{1}{{P + 5}}} \right) + \frac{{P - 6}}{P} + \;\frac{{19}}{{P + 18}} + \;\frac{{19}}{{P - 18}}\;\)


1. \(\frac{{12}}{P} + \frac{{38P}}{{P.P - 324}}\)
2. \(\frac{{12}}{P} + \frac{{19}}{{P.P - 324}}\)
3. \(2 + \frac{{38P}}{{P.P - 324}}\)
4. \(12 + \frac{{38P}}{{P.P - 324}}\)

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Correct Answer - Option 3 : \(2 + \frac{{38P}}{{P.P - 324}}\)

Calculation:

\(\left( {1 + \;\frac{1}{P}} \right)\left( {1 + \;\frac{1}{{P + 1}}} \right)\left( {1 + \;\frac{1}{{P + 2}}} \right)\left( {1 + \;\frac{1}{{P + 3}}} \right)\left( {1 + \;\frac{1}{{P + 4}}} \right)\left( {1 + \;\frac{1}{{P + 5}}} \right) + \frac{{P - 6}}{P} + \;\frac{{19}}{{P + 18}} + \;\frac{{19}}{{P - 18}}\;\)

\(\Rightarrow \left( {\frac{{P + 1}}{P}} \right)\left( {\frac{{P + 2}}{{P + 1}}} \right)\left( {\frac{{P + 3}}{{P + 2}}} \right)\left( {\frac{{P + \;4}}{{P + 3}}} \right)\left( {\frac{{P + 5}}{{P + 4}}} \right)\left( {\frac{{P + \;6}}{{P + 5}}} \right) + \frac{{P - 6}}{P} + \frac{{19\left( {P - 18 + P + 18} \right)}}{{P.P - 18 \times 18}}\) 

\( \Rightarrow \left( {\frac{{P + \;6}}{P}} \right) + \frac{{P - 6}}{P} + \;\frac{{38P}}{{P.P - 324}}\) 

\(\Rightarrow \frac{{2P}}{P} + \frac{{38P}}{{P.P - 324}}\) 

\(\Rightarrow 2 + \frac{{38P}}{{P.P - 324}}\) 

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