12. Find the number of 6-digit positive integers $\overline{a b c d e f}$ satisfying the following two conditions:
(a) Each digit is non-zero.
(b) $a \times b+c \times d+e \times f$ is even.
(1 mark)
求符合以下兩個條件的六位正整數 $\overline{a b c d e f}$ 的數目:
(a) 每位數字皆不等於零。
(b) $a \times b+c \times d+e \times f$ 是偶數。
(1 分)
12. 280616
12. If $a \times b$ is odd, both digits are odd and we have $5 \times 5=25$ choices. If it is even, we have $9 \times 9-5 \times 5=56$ choices (note that both digits cannot be zero). The same is true for the quantities $c \times d$ and $e \times f$.
Now, for condition (b) to hold, either all three quantities $a \times b, c \times d$ and $e \times f$ are even, or exactly two of them are odd. Hence the answer is $56^{3}+3 \times 56 \times 25^{2}=280616$.