What number of lines passing through five points such that no three of them are collinear

Dear Student,

Please find below the solution to the asked query:
$$\begin{gathered} \mathrm{Total} \mathrm{five} \mathrm{points} \\ \mathrm{If} \mathrm{number} \mathrm{of} \mathrm{line} \mathrm{must} \mathrm{passes} \mathrm{with} \mathrm{all} \mathrm{five} \mathrm{points} \mathrm{then} \mathrm{there} \mathrm{is} \mathrm{no} \mathrm{such} \mathrm{line} \mathrm{because} \mathrm{no} \mathrm{three} \mathrm{points} \mathrm{are} \mathrm{collinear} \\ \mathrm{Now} \\ \mathrm{If} \mathrm{lines} \mathrm{passes} \mathrm{through} \mathrm{any} \mathrm{of} \mathrm{the} \mathrm{five} \mathrm{points} \\ \mathrm{Then} \\ \mathrm{Number} \mathrm{of} \mathrm{lines}=\mathrm{Take} \mathrm{any} 2 \mathrm{points} \mathrm{from} 5 \mathrm{points} \mathrm{and} \mathrm{line} \mathrm{passes} \mathrm{through}+\mathrm{line} \mathrm{passes} \mathrm{through} \mathrm{all} \mathrm{five} \mathrm{single} \mathrm{points} \\ \left(\mathrm{we} \mathrm{can} \mathrm{not} \mathrm{add} \mathrm{line} \mathrm{passes} \mathrm{through} 3 \mathrm{or} 4 \mathrm{points} \mathrm{because} \mathrm{no} 3 \mathrm{points} \mathrm{are} \mathrm{collinear}\right) \\ \Rightarrow \mathrm{Number} \mathrm{of} \mathrm{lines}={}^5\mathrm{C}_2+5=10+5=15 \mathrm{lines} \mathrm{Answer} \\ \left({}^5\mathrm{C}_2=10 \mathrm{means} \mathrm{we} \mathrm{can} \mathrm{select} 2 \mathrm{points} \mathrm{from} 5 \mathrm{points} \mathrm{by} 10 \mathrm{ways}\right) \end{gathered}$$

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