All possible four-digit numbers are formed using the

digits $0, 1, 2, 3$  so that no number has repeated digits.

 The number of even numbers among them, is  

In forming even numbers, the position on the right

 can be filled with either O or 2. When O is filled, the

 remaining positions can be filled in 3! ways, and when 

2 is filled, the position on the left can be filled in 2 ways 

(0 cannot be used) and the middle two positions in 2! ways

 (0 can be used)   Req number of numbers =$3!+2+2!=10$