Because we want even numbers, the 1s digit must be 0 or 2.
When the 1s digit is 0, we can freely choose the remaining digits. There are 6 permutations of 3 digits taken 2 at a time (to make a 3-digit number ending in 0), and 6 permutations of 3 digits taken 3 at a time (to make a 4-digit number ending in 0). Thus, there are 12 ways to make a 3- or 4-digit number ending in 0 from the given digits.
When the 1s digit is 2, we need to choose the 100s digit of the 3-digit numbers from the set {1, 3}. The 10s digit can be either of the remaining two digits, for a total of 2*2 = 4 ways to make a 3-digit number ending in 2.
The same restriction applies for the 4-digit numbers ending in 2: The 1000s digit must be from the set {1, 3}, and the remaining digits are free choice. Again, there are 2*2*1 = 4 ways to choose those.
We have a total of 12 numbers ending in 0, and 8 numbers ending in 2, for a total of 20 even numbers greater than 100 made from digits {0, 1, 2, 3}.
The entire list is
120, 130, 210, 230, 310, 320, 1230, 1320, 2130, 2310, 3120, 3210,
102, 132, 302, 312, 1032, 1302, 3012, 3102.
When the 1s digit is 0, we can freely choose the remaining digits. There are 6 permutations of 3 digits taken 2 at a time (to make a 3-digit number ending in 0), and 6 permutations of 3 digits taken 3 at a time (to make a 4-digit number ending in 0). Thus, there are 12 ways to make a 3- or 4-digit number ending in 0 from the given digits.
When the 1s digit is 2, we need to choose the 100s digit of the 3-digit numbers from the set {1, 3}. The 10s digit can be either of the remaining two digits, for a total of 2*2 = 4 ways to make a 3-digit number ending in 2.
The same restriction applies for the 4-digit numbers ending in 2: The 1000s digit must be from the set {1, 3}, and the remaining digits are free choice. Again, there are 2*2*1 = 4 ways to choose those.
We have a total of 12 numbers ending in 0, and 8 numbers ending in 2, for a total of 20 even numbers greater than 100 made from digits {0, 1, 2, 3}.
The entire list is
120, 130, 210, 230, 310, 320, 1230, 1320, 2130, 2310, 3120, 3210,
102, 132, 302, 312, 1032, 1302, 3012, 3102.
