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)

$20$

The number can be of $3$ or $4$ digits.

To find the number of even numbers of three digits, the unit's place must be filled by $0$ and $2$.

Thus the number of even numbers of $3$ digits (having or not having $0$ in hundred${}^{th}$ place) ${=}^2{P}_1{\times }^3{P}_2$

But the number of even numbers of $3$ digits having $0$ at hundred${}^{th}$ place ${=}^2{P}_1$ (as unit's place is naturally filled by $2$ and ten's place by one of $1$ and $3$)

Thus the number of even numbers of $3$ digits ${=}^2{P}_1{\times }^3{P}_2{-}^2{P}_1$.....(1)

Similarly, the numbers of even numbers of $4$ 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 $100$ $=\left({}^2{P}_1{\times }^3{P}_2{-}^2{P}_2\right)+\left({}^2{P}_1{\times }^3{P}_3{-}^2{P}_2\right)=20$