Please help me with 3rd sum
correct answer is 1

Dear student,

Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Note} \mathrm{that} \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{ways} \mathrm{of} \mathrm{ticking} \mathrm{the} \mathrm{answers} \mathrm{in} \mathrm{any} \mathrm{one} \mathrm{attempt} \mathrm{is} \\ 2^4-1=15 \\ \mathrm{The} \mathrm{student} \mathrm{is} \mathrm{taking} \mathrm{chance} \mathrm{at} \mathrm{ticking} \mathrm{the} \mathrm{correct} \mathrm{answer}. \mathrm{It} \mathrm{is} \mathrm{reasonable} \mathrm{to} \\ \mathrm{assume} \mathrm{that} \mathrm{in} \mathrm{order} \mathrm{to} \mathrm{derive} \mathrm{maximum} \mathrm{benefit}, \mathrm{the} \mathrm{five} \mathrm{solutions} \mathrm{which} \mathrm{he} \\ \mathrm{submit} \mathrm{must} \mathrm{be} \mathrm{all} \mathrm{different}. \\ \mathrm{So} \mathrm{the} \mathrm{total} \mathrm{number} \mathrm{of} \mathrm{ways} \mathrm{is} \mathrm{given} \mathrm{by}, \\ \mathrm{n}={ }^{15}{\mathrm{C}}_5 \\ \mathrm{Let} \mathrm{m} \mathrm{be} \mathrm{the} \mathrm{no}. \mathrm{of} \mathrm{ways} \mathrm{in} \mathrm{which} \mathrm{the} \mathrm{correct} \mathrm{solution} \mathrm{is} \mathrm{excluded}. \\ \mathrm{Then} \mathrm{we} \mathrm{have}, \\ \mathrm{m}={ }^{14}{\mathrm{C}}_5 \\ \mathrm{Hence} \mathrm{the} \mathrm{required} \mathrm{probability} \mathrm{is}, \\ \mathrm{p}=1-\frac{{}^{14}{\mathrm{C}}_5}{{}^{15}{\mathrm{C}}_5} \\ =1-\frac{2002}{3003} \\ =\frac{3003-2002}{3003} \\ =\frac{1001}{3003} \\ =\frac{1}{3} \\ \mathrm{Therefore} \mathrm{the} \mathbf{value}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{3}\mathbf{p} \mathrm{is} , \\ 3\mathrm{p}= 3\times \frac{1}{3}=\mathbf{1} \end{gathered}$$

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