Question

# The number of positive numbers less than $$1000$$ and divisible by $$5$$ ( no digits being replaced) is

A
150
B
154
C
166
D
None of these

Solution

## The correct option is B $$154$$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 $$=1$$$$(\because$$ 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 $$\displaystyle =^{ 2 }{ { P }_{ 1 } }\times ^{ 9 }{ { P }_{ 1 } }-1$$,$$(\because$$unit's place can be filled by either $$0$$ or $$5$$ in first category and only by $$5$$ in the second category$$)$$$$=2\times 9-1=17$$.The number of numbers 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 $$5=$$ number of numbers of three digits divisible by $$5$$ and having $$0$$ in hundred's place $$\displaystyle =^{ 2 }{ { P }_{ 1 } }\times ^{ 9 }{ { P }_{ 2 } }\times ^{ 8 }{ { P }_{ 1 } }\times 1=2\times 9\times 8-8=136.$$$$\therefore$$ required number of numbers $$=1+17+136=154$$     Mathematics

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