A question paper consist of two part A and B,part A has 6 questions with one alternate each and part B has 8 questions.Find the number of ways by which one can attempt the paper when at leastone question must be attended from each part.

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$$\begin{gathered} \mathrm{For} \mathrm{section} \mathrm{A} \\ \mathrm{In} \mathrm{each} \mathrm{question} \mathrm{you} \mathrm{have} 3 \mathrm{options}: \\ \mathrm{You} \mathrm{can} \mathrm{answer} \mathrm{the} \mathrm{question} \mathrm{the} \mathrm{question}. \\ \mathrm{You} \mathrm{can} \mathrm{answer} \mathrm{its} \mathrm{alternative}. \\ \mathrm{You} \mathrm{can} \mathrm{skip} \mathrm{the} \mathrm{question}. \\ \mathrm{There} \mathrm{are} 6 \mathrm{questions}, \mathrm{hence} \\ \mathrm{Number} \mathrm{of} \mathrm{ways}=3\times 3\times ... 6 \mathrm{times}=3^6 \\ \mathrm{But} \mathrm{in} \mathrm{above} \mathrm{numbers} \mathrm{of} \mathrm{ways} \mathrm{there} \mathrm{is} \mathrm{one} \mathrm{case} \mathrm{when} \mathrm{you} \mathrm{have} \mathrm{skipped} \mathrm{all} \\ \mathrm{questions} \mathrm{and} \mathrm{as} \mathrm{doing} \mathrm{atleast} \mathrm{one} \mathrm{question} \mathrm{is} \mathrm{necessay}, \mathrm{hence} \\ \mathrm{Favorable} \mathrm{number} \mathrm{of} \mathrm{ways} \mathrm{for} \mathrm{section} \mathrm{A}=3^6-1=\mathbf{728} \\ \mathrm{Similarly} \mathrm{for} \mathrm{section} \mathrm{B} \mathrm{which} \mathrm{has} 8 \mathrm{questions}: \\ \mathrm{In} \mathrm{each} \mathrm{question} \mathrm{you} \mathrm{have} 2 \mathrm{options}: \\ \mathrm{You} \mathrm{can} \mathrm{answer} \mathrm{the} \mathrm{question} \mathrm{the} \mathrm{question}. \\ \mathrm{You} \mathrm{can} \mathrm{skip} \mathrm{the} \mathrm{question}. \\ \mathrm{Hence} \\ \mathrm{Number} \mathrm{of} \mathrm{ways}=2\times 2\times ..... 8 \mathrm{times}=2^8 \\ \mathrm{But} \mathrm{in} \mathrm{above} \mathrm{numbers} \mathrm{of} \mathrm{ways} \mathrm{there} \mathrm{is} \mathrm{one} \mathrm{case} \mathrm{when} \mathrm{you} \mathrm{have} \mathrm{skipped} \mathrm{all} \\ \mathrm{questions} \mathrm{and} \mathrm{as} \mathrm{doing} \mathrm{atleast} \mathrm{one} \mathrm{question} \mathrm{is} \mathrm{necessay}, \mathrm{hence} \\ \mathrm{Favorable} \mathrm{number} \mathrm{of} \mathrm{ways} \mathrm{for} \mathrm{section} \mathrm{B}=2^6-1=\mathbf{255} \\ \mathrm{Hence} \mathrm{required} \mathrm{number} \mathrm{of} \mathrm{ways}=728\times 255=\mathbf{185640}\mathbf{ }\left(\mathbf{Answer}\right) \end{gathered}$$

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