First slide

Permutations

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 three digits $2, 5$ and $7 .$ The smallest value of $n$ for which this is possible is

Moderate

A

6
B

7
C

8
D

9

Solution

The number of n-digit distinct numbers that can be formed using digits $2, 5$ and 7 is $3^{n} .$
We have to find $n$ , so that $3^{n} \geq 900$
$\Rightarrow 3^{n - 2} \geq 100 \Rightarrow n - 2 \geq 5 \Rightarrow n \geq 7$
Thus, the smallest value of $n$ is $7$

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