How many three digit numbers are there which have exactly one of their digits as 6.(This was the question given ,but does it mean the numbers can be repeated or not )

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Yes} \mathrm{digits} \mathrm{can} \mathrm{be} \mathrm{repeated}. \mathrm{Now} \mathrm{let} \mathrm{us} \mathrm{solve} \mathrm{the} \mathrm{problem}. \\ \mathbf{Case}\left(\mathbf{i}\right)\mathbf{ }\mathbf{When}\mathbf{ }\mathbf{6}\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{first}\mathbf{ }\mathbf{digit}\mathbf{.} \\ \mathrm{Number} \mathrm{of} \mathrm{ways} \mathrm{to} \mathrm{fill} \mathrm{the} \mathrm{first} \mathrm{digit}=1 \left(\mathrm{i}.\mathrm{e}. 6\right) \\ \mathrm{Number} \mathrm{of} \mathrm{ways} \mathrm{to} \mathrm{fill} \mathrm{the} \mathrm{second} \mathrm{digit}=9 \left(\mathrm{All} \mathrm{digits} \mathrm{except} 6\right) \\ \mathrm{Number} \mathrm{of} \mathrm{ways} \mathrm{to} \mathrm{fill} \mathrm{the} \mathrm{third} \mathrm{digit}=9 \left(\mathrm{All} \mathrm{digits} \mathrm{except} 6\right) \\ \Rightarrow \mathrm{Total} \mathrm{cases}=9\times 9\times 1=81 \\ \mathbf{Case}\left(\mathbf{ii}\right)\mathbf{ }\mathbf{When}\mathbf{ }\mathbf{6}\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{second}\mathbf{ }\mathbf{digit}\mathbf{.} \\ \mathrm{Number} \mathrm{of} \mathrm{ways} \mathrm{to} \mathrm{fill} \mathrm{the} \mathrm{first} \mathrm{digit}=8\left(\mathrm{All} \mathrm{digits} \mathrm{except} 6 \mathrm{and} 0\right) \\ \mathrm{Number} \mathrm{of} \mathrm{ways} \mathrm{to} \mathrm{fill} \mathrm{the} \mathrm{second} \mathrm{digit}=1 \left(\mathrm{i}.\mathrm{e}. 6\right) \\ \mathrm{Number} \mathrm{of} \mathrm{ways} \mathrm{to} \mathrm{fill} \mathrm{the} \mathrm{third} \mathrm{digit}=9 \left(\mathrm{All} \mathrm{digits} \mathrm{except} 6\right) \\ \Rightarrow \mathrm{Total} \mathrm{cases}=9\times 8\times 1=72 \\ \mathbf{Case}\left(\mathbf{iii}\right)\mathbf{ }\mathbf{When}\mathbf{ }\mathbf{6}\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{third}\mathbf{ }\mathbf{digit}\mathbf{.} \\ \mathrm{Number} \mathrm{of} \mathrm{ways} \mathrm{to} \mathrm{fill} \mathrm{the} \mathrm{first} \mathrm{digit}=8\left(\mathrm{All} \mathrm{digits} \mathrm{except} 6 \mathrm{and} 0\right) \\ \mathrm{Number} \mathrm{of} \mathrm{ways} \mathrm{to} \mathrm{fill} \mathrm{the} \mathrm{second} \mathrm{digit}=9 \left(\mathrm{All} \mathrm{digits} \mathrm{except} 6\right) \\ \mathrm{Number} \mathrm{of} \mathrm{ways} \mathrm{to} \mathrm{fill} \mathrm{the} \mathrm{third} \mathrm{digit}=1 \left(\mathrm{i}.\mathrm{e}. 6\right) \\ \Rightarrow \mathrm{Total} \mathrm{cases}=9\times 8\times 1=72 \\ \mathbf{Total}\mathbf{ }\mathbf{3}\mathbf{ }\mathbf{digit}\mathbf{ }\mathbf{number}\mathbf{ }\mathbf{with}\mathbf{ }\mathbf{exactly}\mathbf{ }\mathbf{one}\mathbf{ }\mathbf{6}\mathbf{=}\mathbf{81}\mathbf{+}\mathbf{72}\mathbf{+}\mathbf{72}\mathbf{=}\mathbf{225}\mathbf{ }\left(\mathbf{Answer}\right) \end{gathered}$$

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