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  • An n-digit number is a positive number with exactly n digits. Seven hundred distinct n-digit numbers are to be formed using only the four digits 0, 5, 6 and 7. The smallest value of n for which thi

An n-digit number is a positive number with exactly n digits. Seven hundred distinct n-digit numbers are to be formed using only the four digits 0, 5, 6 and 7. The smallest value of n for which this is possible is _______.

 

Option: 1

8


Option: 2

5


Option: 3

7


Option: 4

1


Answers (1)

To find the smallest value of n for which 700 distinct n-digit numbers can be formed using only the digits 0, 5, 6, and 7, we need to determine the maximum number of distinct numbers that can be formed with a given number of digits.

Since we have four digits to choose from (0, 5, 6, and 7), and we want to form distinct numbers, each digit can only be used once in each number. Therefore, the maximum number of distinct numbers that can be formed with n digits is given by 4^n (4 raised to the power of n).

We want to find the smallest value of n for which 4^n is greater than or equal to 700. Let's calculate the values of 4^n for increasing values of n until we find a value that satisfies this condition:

\begin{aligned} & 4^1=4 \\ & 4^2=16 \\ & 4^3=64 \\ & 4^4=256 \\ & 4^5=1024 \end{aligned}

We see that 4^5=1024  is the smallest power of 4 that is greater than 700 . Therefore, we need at least 5 digits to form 700 distinct numbers using the digits 0,5,6, and 7 .

However, since we want exactly 700 distinct numbers, we need to check if we can form exactly 700 numbers with 5 digits. If we have 5 digits, the maximum number of distinct numbers we can form is 4^5=1024 which is greater than 700 .

Therefore, the smallest value of n for which it is possible to form exactly 700 distinct n-digit numbers using only the digits 0, 5, 6, and 7 is 5.

 

Posted by

Sumit Saini

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