How many numbers between 100 and 1000 are such that exactly one of the digits is 6?

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225

425

625

825

answer is A.

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Detailed Solution

Concept- To solve this question, first we need to calculate the count of numbers that have 6 at the hundred’s place, the count of numbers that have 6 at the ten’s place, and the count of numbers that have 6 at the unit’s place between 100 and 1000. Finally, we will add these counts calculated from the above-mentioned three cases to obtain our desired answer.
Given, we need to count the numbers between 100 and 1000 which have exactly one of the digits as 6. Since, All the numbers between 100 and 1000 are three digit numbers, therefore, exactly one of the three digits must be 6.
We will consider three cases to find our required answer, the cases being:
Case 1: The number 6 is at the hundred's place.
The three digit number will have only one 6 .
Therefore, the remaining digits are 0,1,2,3,4,5,7,8,9.
In the ten's place, any of the remaining 9 numbers (except 6 ) can be placed.
In the unit's place, any of the remaining 9 numbers (except 6 ) can be placed.
Therefore, we get,
Number of three digit numbers with exactly one 6 (at the hundred's place) =(9

\times

9)=81.
Case 2: The number 6 is at the ten's place.
The three digit number will have only one 6 .
Therefore, the remaining digits are 0,1,2,3,4,5,7,8,9.
We know that a three digit number cannot have the digit 0 in the hundred's place.
Thus, in the hundreds place, any of the remaining 8 numbers (except 6 and 0 ) can be placed.
In the unit's place, any of the remaining 9 numbers (except 6 ) can be placed.
Therefore, we get,
Number of three digit numbers with exactly one 6 (at the ten's place )=(8

\times

9)=72.
Case 3: The number 6 is at the unit's place.
The three digit number will have only one 6 .
Therefore, the remaining digits are $0,1,2,3,4,5,7,8,9$.
We know that a three digit number cannot have the digit 0 in the hundred's place.
Thus, in the hundreds place, any of the remaining 8 numbers (except 6 and 0 ) can be placed.
In the teri's place, any of the remaining 9 numbers (except 6 ) can be placed.
Therefore, we get,
Number of three digit numbers with exactly one 6 (at the unit's place ) =(8