Question

The number of even numbers greater than $100$ that can be formed by the digits $0,1,2,3$ (no digit being repeated) is

• A. 20
• B. 30
• C. 40
• D. None of these

Answer 1

Answer: (A)

The numbers are of three or four digits.
To find the number of even numbers of three digits
The unit's place must be filled by $0$ or $2$ .
$\therefore$ The number of even numbers of three digits
(having or not having $0$ in hundred's place)
$={}^2{P}_1\times {}^3{P}_2$.
But the number of even numbers of three digits having $0$ in hundred's place $={}^2{P}_1$
($\because$ unit's place is naturally filled by $2$ and ten's place by one of $1$ and $3$ ).
$\therefore$ The number of even numbers of three digits
$={}^2{P}_1\times {}^3{P}_2-{}^2{P}_1(1)$
Similarly, the number of even numbers of four digits
$={}^2{P}_1\times {}^3{P}_3-{}^2{P}_2(2)$
Adding (1) and (2), we get the number of even numbers greater than $100$
$=\left({}^2{P}_1\times {}^3{P}_2-{}^2{P}_1\right)+\left({}^2{P}_1\times {}^3{P}_3-{}^2{P}_2\right)$
$=2\times 3\times 2-2+2\times 3\times 2\times 1-2\times 1=20$