The number of positive numbers less than $$1000$$ and divisible by $$5$$ (no$$ digit being $$repeated) is

Question

Infinity Learn · Accepted Answer

Here, the available digits are $$0$$, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$.The numbers can be of one, two or three digits and in each of them unit’s place must have $$0$$ or $$5$$ as they must be divisible by $$5$$.The number of numbers of one digit $$=$$ $$1$$ $$($$∵ $$5$$ is the only $$number)$$.The number of numbers of two digits divisible by $$5$$ $$=$$ number of all the numbers of two digits divisible by $$5$$ – number of numbers of two digits divisible by $$5$$ and having $$0$$ in ten’s place $$=2P1$$×$$9P1$$−$$1$$,$$($$∵ unit’s place can be filled by either $$0$$ or $$5$$ in first category and only by $$5$$ in the second $$category)=$$ $$2$$ × $$9$$ – $$1$$ $$=$$ $$17$$.The number of numbers of three digits divisible by $$5$$ $$=$$ number of all the numbers of three digits divisible by $$5$$ $$=$$ number of numbers of three digits divisible by $$5$$ and having $$0$$ in hundred’s $$place=2P1$$×$$9P2$$−$$8P1$$×$$1=2$$×$$9$$×$$8$$−$$8=136$$.∴ Required number of $$numbers=$$ $$1$$ $$+$$ $$17$$ $$+$$ $$136$$ $$=$$ $$154$$.

The number of positive numbers less than 1000 and divisible by 5 (no digit being repeated) is

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