Count the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided

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Q. Count the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digits are repeated.

Tamil Nadu DgeTamil Nadu HSC First Year (DGE) Board 2019Subjective· 2mImportance★★★★★

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Fix the first digit as 6 (required by the range), let the last digit be 0 or 5 (required by divisibility by 5), and arrange the two middle digits from the remaining 8 available digits without repetition: $2\times(8\times7)=112$ .

A number greater than 6000 and less than 7000 is a 4-digit number of the form $6\,\_\,\_\,\_$ — the thousands digit is fixed as 6.

For divisibility by 5, the units digit must be 0 or 5.

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2025Set ANNUAL1 markMCQ
$Q. The number of 5 digit numbers, all digits of which are odd is:$
(a) $5^6$
(b) $25$
(c) $625$
(d) $5^5$
›Reveal solutionSolution

Each of the 5 digit-positions is filled independently from the 5 odd digits.

The odd digits available are $\{1,3,5,7,9\}$ , a set of 5 digits. Since none of them is $0$ , there is no restriction on the leading digit either. Each of the 5 positions of the number can be filled in 5 ways, independ …

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CBSE 2022Set ANNUAL1 markMCQ
$Q. The number of 5 digit numbers all digits of which are odd is:$
(a) $5^6$
(b) 25
(c) 625
(d) $5^5$
›Reveal solutionSolution

With 5 choices of odd digit for each of the 5 positions (no restriction, since all odd digits are already non-zero), the count is $5^5$ .

The odd digits are $1,3,5,7,9$ -- exactly 5 of them.

A 5-digit number has 5 positions, and unlike a general 5-digit number problem, there's no separate restriction on the leading digit here, because none of the allowed digits (odd digits) is zero anyway.

…

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CBSE 2019Set ANNUAL1 markMCQ
$Q. The number of five digit numbers in which all digits are even, is:$
(a) $4\times 5^4$
(b) $4\times 5^5$
(c) $5^5$
(d) $5\times 5$
›Reveal solutionSolution

With digits restricted to $\{0,2,4,6,8\}$ , the first digit excludes 0 (4 choices) and the remaining 4 digits are unrestricted (5 choices each), giving $4\times 5^4$ .

The even digits available are $0, 2, 4, 6, 8$ — 5 digits in total, and repetition is allowed (the question only restricts which digits may be used, not that they be distinct).

For a 5-digit number, the first (leftmost) digit cannot be 0, so it has $5-1=4$ choices.

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Q.Count the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digits are repeated.

Concept understanding — Fundamental Principles of Counting

The Fundamental Principle of Counting

The Intuition: A Fork in the Road

The Precise Statement

A Simple Example

Q.Count the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digits are repeated.

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