The correct option is **B**

$7$

**TFinding the smallest value of n:**

Given,

Three digit number $2,5and7$ with repetition, each place of n digit number can be chosen in $3$ ways.

See AlsoThe SIREN and SIRET NumbersFAQ - Registration - Social security numberDefinition - Internal classification number / NIC (SIRENE)France INSEE NumberHence, the total number of n-digit numbers$=3\times 3\times 3...ntimes={3}^{\mathrm{n}}$

According to given condition

$\begin{array}{rcl}{3}^{\mathrm{n}}& \ge & 900\\ & \Rightarrow & {3}^{\mathrm{n}-2}\ge 100\\ & & \end{array}$

Now, here we need to take the value of n which satisfies the left-hand side of the equation to the right-hand side.

Taking, $n=6$

$\begin{array}{rcl}& \Rightarrow & {3}^{4}=81\\ 81& \le & 100\end{array}$not satisfy the equation

**Hence option (A) is incorrect **

Therefore,

$\begin{array}{rcl}& \Rightarrow & n-2=5\\ n& =& 7\end{array}$

Taking,

$\begin{array}{rcl}& \Rightarrow & {3}^{5}=243\\ 243& \ge & 100\end{array}$which satisfies our equation.

**Hence option **$\left(B\right)$** is correct.**

Taking, $n=8$

$\begin{array}{rcl}& \Rightarrow & {3}^{6}=729\\ 729& \ge & 100\end{array}$which satisfy our equation but $8\ge 7$

**Hence, option (C) is incorrect.**

Taking, $n=9$

$\begin{array}{rcl}& \Rightarrow & {3}^{7}=2187\\ 2187& \ge & 100\end{array}$which also satisfy the equation, but $9\ge 7$

**Hence option **$\left(B\right)$** is correct.**

The correct option is **B**

$7$

**TFinding the smallest value of n:**

Given,

Three digit number $2,5and7$ with repetition, each place of n digit number can be chosen in $3$ ways.

Hence, the total number of n-digit numbers$=3\times 3\times 3...ntimes={3}^{\mathrm{n}}$

According to given condition

$\begin{array}{rcl}{3}^{\mathrm{n}}& \ge & 900\\ & \Rightarrow & {3}^{\mathrm{n}-2}\ge 100\\ & & \end{array}$

Now, here we need to take the value of n which satisfies the left-hand side of the equation to the right-hand side.

Taking, $n=6$

$\begin{array}{rcl}& \Rightarrow & {3}^{4}=81\\ 81& \le & 100\end{array}$not satisfy the equation

**Hence option (A) is incorrect **

Therefore,

$\begin{array}{rcl}& \Rightarrow & n-2=5\\ n& =& 7\end{array}$

Taking,

$\begin{array}{rcl}& \Rightarrow & {3}^{5}=243\\ 243& \ge & 100\end{array}$which satisfies our equation.

**Hence option **$\left(B\right)$** is correct.**

Taking, $n=8$

$\begin{array}{rcl}& \Rightarrow & {3}^{6}=729\\ 729& \ge & 100\end{array}$which satisfy our equation but $8\ge 7$

**Hence, option (C) is incorrect.**

Taking, $n=9$

$\begin{array}{rcl}& \Rightarrow & {3}^{7}=2187\\ 2187& \ge & 100\end{array}$which also satisfy the equation, but $9\ge 7$

**Hence option **$\left(B\right)$** is correct.**