Permutations and Combinations 1 Question 6
6. 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
$(1998,2 M)$
(a) 6
(b) 7
(c) 8
(d) 9
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Answer:
Correct Answer: 6. (c)
Solution:
- Distinct $n$-digit numbers which can be formed using digits 2,5 and 7 are $3^{n}$. We have to find $n$, so that
$$ \begin{aligned} & & 3^{n} & \geq 900 \\ \Rightarrow & & 3^{n-2} & \geq 100 \\ \Rightarrow & & n-2 & \geq 5 \end{aligned} $$
$\Rightarrow \quad n \geq 7$, so the least value of $n$ is 7 .
