12. Determine the number of quadruples of integers (not necessarily distinct) between 1 and 12 (inclusive) that satisfy all the following conditions:

- the sum of the first two numbers is even
- the sum of the first three numbers is a multiple of 3
- the sum of the four numbers is a multiple of 4.

(Two quadruples that differ even in the order of the addends are to be considered distinct).
12. The answer is 864. For each choice of the first number, the second number can be chosen in 6 ways to satisfy the first condition; still, for each possible sum of the first 2 numbers, there are 4 possible choices for the third so that the second condition is satisfied; finally, for each possible choice of the first 3 numbers, there are 3 possibilities for the fourth. Thus, the total number of possibilities is $12 \cdot 6 \cdot 4 \cdot 3=864$.