Question

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

Answer 1

The number of sequences of 6 digits can be found by considering the first digit as odd and then the second digit as even or vice versa.

Step-by-step explanation:

The number of sequences of 6 digits can be found by considering the first digit as odd and then the second digit as even or vice versa.

When the first digit is odd, there are 5 choices for the first digit and 5 choices for the second digit. This gives us a total of 5 * 5 = 25 choices.

Similarly, when the first digit is even, there are 5 choices for the first digit and 4 choices for the second digit, resulting in a total of 5 * 4 = 20 choices. Therefore, the total number of sequences is 25 + 20 = 45.