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Permutations and Combination — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsPermutations and Combination4 Marks
Question
How many numbers between 100 and 1000 have the digit 7 exactly once?
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Answer

Numbers 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. When 7 is in unit’s place: Unit’s place digit is 7. ∴ it can be selected in 1 way only. 10’s place digit can be selected in 9 ways. 100’s place digit can be selected in 8 ways. Total numbers which have 7 in unit’s place = 1 × 9 × 8 = 72. When 7 is in 10’s place: Unit’s place digit can be selected in 9 ways. 10’s place digit is 7. ∴ it can be selected in 1 way only. 100’s place digit can be selected in 8 ways. ∴ A total number of numbers which have 7 in 10’s place = 9 × 1 × 8 = 72. When 7 is in 100’s place: Unit’s place digit can be selected in 9 ways. 10’s place digit can be selected in 9 ways. 100’s place digit is 7. ∴ it can be selected in 1 way only. ∴ Total numbers which have 7 in 100’s place = 9 × 9 × 1 = 81. ∴ Total numbers between 100 and 1000 having digit 7 exactly once = 72 + 72 + 81 = 225

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