Probability of sequences of dice rolls

Question

I would really appreciate help with a calculation of probabilities.

Let's say I have a die with six sides. I roll the die 7 times and get the following sequence: 5, 1, 2, 2, 4, 5, 1

Given some random other sequence (e.g. 6, 1, 3, 2, 5, 5, 5), what is the probability that there are exactly two numbers in the same position? Exactly three numbers in the same position? And so on.

For example, with the above sequences there are three numbers in the same position: 1 in second position, 2 in fourth position and 5 in sixth position.

Answer 1

Hint:

Without Loss of Generality, the first sequence is
1, 1, 1, 1, 1, 1, 1.

Answer 2

Hint:

We have a reference sequence to match. In how many ways, can there be a match in exactly two positions? There are $\displaystyle {7 \choose 2}$ ways to choose two positions. Each position matches with a probability of $\displaystyle \frac{1}{6}$ and is different with a probability of $\displaystyle \frac{5}{6}$.