Question

An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2,5 and 7. The smallest value of n for which this is possible, is

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

The Correct Option is B

Distinct n-digit numbers which can be formed using digits 2,5 and 7 are 3$^n$ We have to find n, so that $3^n \ge 900$
$\Rightarrow$ 3$^{n-2}\ge 100$
$\Rightarrow \, \, n-2 \ge 5$
$\Rightarrow $ n $\ge$ 7, so the least value of n is 7

Answer 2

Given, that we can use the digits 2, 5, and 7 with repetition, each place in an n-digit number can be chosen in 3 different ways.
So, the total number of n-digit numbers = \(3\times3\times3\times.....n\;=\;3^n\)
According to question, \(3^n\geq900\)
Let's simplify
\(3^n\geq 3^2\times100\)
\(3^{n-2}\geq100\)

Let n=6
\(3^{6-2}\geq100\)
\(3^{4}\geq100\)
\(81\geq100\), which does not satisfy the condition

let n=7
\(3^{7-2}\geq100\)
\(3^{5}\geq100\)
\(243\geq100\), which is satisfy the condition and also the smallest number from 6,7,8 and 9

So, the correct option is (B): 7.