Choose the 1st digit any of these 6 ways {1,2,3,4,5,6}
Choose the 2nd digit any of these 6 ways {1,2,3,4,5,6}
Choose the 3rd digit any of these 6 ways {1,2,3,4,5,6}
Choose the 4th digit any of these 6 ways {1,2,3,4,5,6}
Choose the 5th digit any of these 3 ways {2,4,6}

That's 6×6×6×6×3 = 3888 ways.

We must choose the last digit first to make sure it's even:

Choose the 5th digit any of these 3 ways {2,4,6}.
Choose the 1st digit any of the 5 ways remaining that weren't used in the 5th
digit.
Choose the 2nd digit any of the 4 ways remaining that weren't used in the 5th
or 1st digits.
Choose the 3rd digit any of the 3 ways remaining that weren't used in the 5th,
1st, or 2nd digits.
Choose the 3rd digit any of the 2 ways remaining that weren't used in the 5th,
1st, 2nd, or 3rd digits.

That's 3×5×4×3×2 = 360 ways.

Edwin