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Question
How many numbers between 100 and 1000 have the digit 7 exactly once?
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Solution
Between 100 and 1000 we have all 3-digit numbers and digits used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Digit 7 can be placed either in unit's place or in ten's place or in hundred's place.
(i) Digit 7 in unit's place:
Unit's place can be filled in only in 1 way.
Ten's place can be filled in by remaining 9 digits in 9 different ways.
Hundred's place can be filled in by (except '0') remaining 8 digits in 8 different ways.
∴ the total numbers between 100 and 1000 with 7 at unit's place
= 8 × 9 × 1
= 72
(ii) Digit 7 in ten's place:
Unit's place can be filled in by remaining 9 digits (except 7) in 9 different ways.
Ten's place can be filled in by 7 in 1 way only.
Hundred's place can be filled in by remaining 8 digits (except 0 and 7) in 8 different ways .
∴ the total numbers between 100 and 1000 with 7 at ten's place
= 8 × 1 × 9
= 72
(iii) Digit 7 in hundred's place:
Unit's place can be filled in by remaining 9 digits (except 7) in 9 different ways.
Ten's place can be filled in by remaining 9 digits (except 7) in 9 different ways.
Hundred's place can be filled in by digit 7 in only 1 way.
∴ the total numbers between 100 and 1000 with 7 at hundred's place
= 1 × 9 × 9
= 81
Hence, there are 72 + 72 + 81 = 225 numbers between 100 and 1000 such that exactly one of the digits is 7.
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