How many 10-bit binary sequences begin and end with the same bit?

Question

Can you please teach me how to solve this? There are other similar questions that I have, but I don't understand how to solve them. Another question is: How many 10-bit binary sequences contain an odd number of 1s?

Answer 1

1. They can either end with $0$'s or $1$'s (2 choices). The remaining 8 bits can be either $1$ or $0$ ($2^8$ choices), so you have $2^9$ different possibilities.

2. Exactly half of them. The total amount of possibilities is $2^{10}$, so half of this is $\frac{2^{10}}{2} = 2^9$.