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Question

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


A
150
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B
154
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C
166
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D
None of these
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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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