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How many numbers are there between 100 and 1000 such that at least one of their digits is 7? 

(a) 900 

(b) 648 

(c) 729 

(d) 252

1 Answer

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(b) 648

A number between 100 and 1000 has 3-digits 

∴ Total number of 3-digit numbers having at least one of their digits as 7 = (Total number of 3-digit numbers) – (Total number of 3-digit numbers in which 7 does not appear at all) 

(a) Total number of 3-digit numbers: 

Clearly repetition of digits is allowed. 

The hundred’s place can be filled in 9 ways, i.e., using any of the digits 1 to 9 excluding zero. 

The ten’s and the unit’s place can be filled in 10 ways using all the digits from 0 – 9 in each place. 

∴ Total number of 3-digit numbers = 9 × 10 × 10 = 900 

(b) Total number of 3-digit numbers in which 7 does not appear at all: 

Here, the digits to be used are 0, 1, 2, 3, 4, 5, 6, 8, 9 i.e., 9 in number. 

Now, the hundred’s place can be filled in 8 ways (excluding 0), the tens’ and ones’ place can be filled in 9 ways each. 

∴ Total number of 3-digit numbers in which 7 does not appear, = 8 × 9 × 9 = 648

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