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Question
How many numbers between 100 and 1000 have the digit 7 exactly once?
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Solution
A number between 100 and 1000 are 3-digit numbers.
A 3-digit number is to be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, where exactly one of the digits is 7.
Let us consider the three cases separately.
Case (I): the digit 7 is in the unit’s place.
The ten’s place is filled by one digit from 0 to excluding 7 in 9 ways.
Here, there are 8 × 9 × 1 = 72 three digit numbers with the required condition.
Case (II): The digit 7 is in the ten’s place.
Unit’s place can be filled by digit from 0 to 9 excluding 7 in 9 ways.
Zero is not allowed at a hundred’s place.
Hundred's place can be filled by digit from 1 to 9 excluding 7 in 8 ways.
The hundred’s place can be filled in by any digit from 1 to 9 excluding 7 in 8 ways.
Here, there will be 8 × 1 × 9 = 72
3-digit numbers with the required condition
Case (III): The digit 7 is in the hundred’s place.
Then, there are 1 × 9 × 9 = 81
3-digit numbers with the required condition.
Hence, the numbers between 100 and 1000 having the digit 7 exactly once are 72 + 72 + 81 = 225.
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