Question

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

• A. 150
• B. 154
• C. 166
• D. None of these

Answer 1

Answer: (B)

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 $=1$
( $\because 5$ 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 $={}^2{P}_1\times {}^9{P}_1-1$,
($\because$ 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 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
$={}^2{P}_1\times {}^9{P}_2-{}^8{P}_1\times 1=2\times 9\times 8-8=136$
$\therefore$ Required number of numbers
$=1+17+136=154$.