There is only one condition required for a vector to be even: an even number of entries are odd.

For a solid example, consider vectors with four entries, . The total number of those vectors is . Then, any even vector can have 0, 2, or 4 odd entries. Here are the cases:

0 odd entries: arrangements

2 odd entries: arrangements

4 odd entries: arrangements

But take note that there are some degrees of freedom with each situation. For example, in the case of 0 odd entries, there are no choices for odd entries, but 2 choices for even. Therefore, there are choices of elements. Similarly, for 2 odd entries, there are choices, and for 4 odd entries, there are choices.

In summary, there are

even vectors. This is exactly half of the total number of vectors.

Try to generalize this argument for the case where is arbitrary.