[n-3][n-4][n-5]= 210

find the value of n
....pls explain each and every step

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{As} \mathrm{question} \mathrm{is} \mathrm{from} \mathrm{Permutations} \mathrm{and} \mathrm{Combinations}, \mathrm{hence} \mathrm{I} \mathrm{am} \mathrm{assuming} \mathrm{that} \\ \mathrm{n}>0. \\ \mathrm{We} \mathrm{have}: \\ \left(\mathrm{n}-3\right)\left(\mathrm{n}-4\right)\left(\mathrm{n}-5\right)=210 \\ \mathrm{Now} \mathrm{L}.\mathrm{H}.\mathrm{S}. \mathrm{is} \mathrm{product} \mathrm{of} \mathrm{three} \mathrm{consecutive} \mathrm{numbers}, \mathrm{hence} \mathrm{we} \mathrm{will} \mathrm{try} \mathrm{to} \\ \mathrm{split} \mathrm{R}.\mathrm{H}.\mathrm{S}. \mathrm{in} \mathrm{product} \mathrm{of} \mathrm{three} \mathrm{consecutive} \mathrm{numbers}. \\ 210=21\times 10=7\times 3\times 10=7\times 3\times 5\times 2=7\times 6\times 5 \\ \Rightarrow \left(\mathrm{n}-3\right)\left(\mathrm{n}-4\right)\left(\mathrm{n}-5\right)=7\times 6\times 5 \\ \mathrm{In} \mathrm{L}.\mathrm{H}.\mathrm{S}. \left(\mathrm{n}-3\right) \mathrm{is} \mathrm{the} \mathrm{largest} \mathrm{number} \mathrm{and} \mathrm{on} \mathrm{R}.\mathrm{H}.\mathrm{S}. 7 \mathrm{is} \mathrm{the} \mathrm{largest} \mathrm{number}. \\ \mathrm{n}-3=7 \\ \Rightarrow \mathrm{n}=7+3 \\ \mathbf{\Rightarrow }\mathbf{n}\mathbf{=}\mathbf{10}\mathbf{ }\mathbf{ }\left(\mathbf{Answer}\right) \end{gathered}$$
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