### Instruction: How many sequences of 6 digits $x_1, x_2, \ldots, x_6$ can we form, given the condition that no two adjacent $x_i$ have the same parity? Leading zeroes are allowed.  (Parity means 'odd' or 'even'; so, for example, $x_2$ and $x_3$ cannot both be odd or both be even.).### Response: Let's start with the first digit. It can be any number, so there are 10 choices.
That's right. And the second digit must be the opposite parity of the first digit, so there are 5 choices.
That's correct. And the third digit must be the opposite parity of the second digit, so there are 5 choices.
So after three digits, we have $10 \cdot 5 \cdot 5 = 250$ sequences.
Right. And the fourth digit must be the opposite parity of the third digit, so there are 5 choices.
So after four digits, we have $10 \cdot 5 \cdot 5 \cdot 5 = 1250$ sequences.
Right. And the fifth digit must be the opposite parity of the fourth digit, so there are 5 choices.
So after five digits, we have $10 \cdot 5 \cdot 5 \cdot 5 \cdot 5 = 6250$ sequences.
Right. And the sixth digit must be the opposite parity of the fifth digit, so there are 5 choices.
So in total, we have $10 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 = 31250$ sequences.