Permutations of bit-sequence(discrete math) How many bit-squences with length 8 has 1 as it's first bit and 00 as the two last bits(e.g $1011 1100$)
I thought the solution to this problem would be $1 * 2 * 2 * 2 * 2 * 2 * 1 * 1 = 2^5$, but my teacher proposed the following solution:
amount that starts with 1:  $1 * 2 * 2 * 2 * 2 * 2 * 2 = 2^7$
amount that ends with 00: $2 * 2 * 2 * 2 * 2 * 2 * 1 * 1 = 2^6$
amount that starts with 1 and ends with 00:  $1 * 2 * 2 * 2 * 2 * 2 * 1 * 1 = 2^5$
answer = $2^7 + 2^6 - 2^5 = 160$.

Isn't the amount of bit-squences that starts with 1 and ends with $00$ the answer to the question "How many bit-squences with length 8 has 1 as it's first bit and $00$ as the two last bits"? Isn't the answer simply $2^5$ ?

Yes, the answer is simply $2^5$. Your teacher is calculating some other count.  
OK, your teacher is calculating the count of those sequences which start with 1 or end with 00. And she's using the inclusion exclusion principle for counting them.  
Inclusion-Exclusion Principle
Maybe you misunderstood the problem. You thought there was an and there in the problem statement but it's actually an or.
