1. there are n locks and n matching keys if all the locks and keys are to be perfectly matched find the maximum number of trials required to open a lock?

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{For} \mathrm{the} \mathrm{first} \mathrm{key}, \mathrm{maximum} \mathrm{number} \mathrm{of} \mathrm{trials} \mathrm{needed} \mathrm{is} \mathrm{n}. \mathrm{For} \mathrm{second} \mathrm{key}, \mathrm{it} \mathrm{will} \mathrm{be} \left(\mathrm{n} – 1\right) . \\ \mathrm{In} \mathrm{general}, \mathrm{for} {\mathrm{r}}^{\mathrm{th}} \mathrm{key}, \mathrm{maximum} \mathrm{number} \mathrm{of} \mathrm{trials} \mathrm{needed} \mathrm{are} (\mathrm{n} – \mathrm{r} + 1), \mathrm{then} \mathrm{maximum} \mathrm{number} \\ \mathrm{of} \mathrm{trial}s \mathrm{are} \mathrm{T}=\mathrm{n}+\left(\mathrm{n}-1\right)+\left(\mathrm{n}-2\right)+.......+1 \\ \mathrm{Sum} \mathrm{of} \mathrm{n} \mathrm{terms} \mathrm{of} \mathrm{an} \mathrm{A}.\mathrm{P}.=\frac{\mathrm{n}}{2}\left(\mathrm{First} \mathrm{Term}+\mathrm{Last} \mathrm{Term}\right) \\ \Rightarrow \mathrm{T}=\frac{\mathrm{n}}{2}\left(\mathrm{n}+1\right) \\ \Rightarrow \mathrm{T}=\frac{\mathrm{n}\left(\mathrm{n}+1\right)}{2}{=}^{\mathrm{n}+1}{\mathrm{C}}_2 \\ \mathbf{Hence}\mathbf{ }\mathbf{maximum}\mathbf{ }\mathbf{number}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{trials}\mathbf{ }\mathbf{needed}\mathbf{ }\mathbf{are}{\mathbf{ }}^{\mathbf{n}\mathbf{+}\mathbf{1}}{\mathbf{C}}_{\mathbf{2}}\mathbf{.} \end{gathered}$$

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