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

Answer 1

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 ${=}^2{P}_1{\times }^9{P}_1-1$,

$(\because$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

${=}^2{P}_1{\times }^9{P}_2{\times }^8{P}_1\times 1=2\times 9\times 8-8=136.$

$\therefore$ required number of numbers

$=1+17+136=154$