a) We solve this problem by first finding the probability of NONE of the $17$ people having the same birthday.
Person $1$ has a birthday (let's call it $D_1$). Then Person $2$ must have a birthday other than $D_1$ (let's call it $D_2$), which leaves him $365-1 = 364$ options. Person $3$ must have a birthday other than $D_1$ and $D_2$ (let's call it $D_3$), which leaves him $365-2 = 363$ options. This goes on until you reach $D_{17}$.
Because there are $365^{17}$ arrangements of birthday possible, and you have $365 \cdot 364 \cdot 363 \cdots 349$ options, the answer is $\displaystyle \frac{365 \cdot 364 \cdot 363 \cdots 349}{365^{17}}$.
Note that we're not finished yet, however, because we found the probability that NONE of them have the same birthday when we really want the probability that AT LEAST TWO have the same birthday. Fortunately, the math is simple - we just subtract the value we found from $1$. So the final answer is $$ 1 - \frac{365 \cdot 364 \cdot 363 \cdots 349}{365^{17}}.$$
b) We can choose two people among the $17$ to have the birthday of January $1$st, and we don't care when the others have their birthdays as long as it's not January $1$st.
We have to choose the two people. Because order doesn't matter, this will be a combination - namely, $\dbinom{17}{2}$. The probability that both of them have the birthday of January $1$st is $\displaystyle \left(\frac{1}{365}\right)^2$.
We then have to assign a birthday to each of the others (N.B.: They can have the same birthdays), which means the probability will be $\displaystyle \left(\frac{364}{365}\right)^{15}$).
Multiplying all of these together, we get
$$\dbinom{17}{2} \left(\frac{1}{365}\right)^2 \left(\frac{364}{365}\right)^{15}.$$
Best Answer
It certainly changes things. Say $n=5$, then we know at least one of the possible pairs have the same birthday. There are $10$ pairs, and each is equally likely to have the same birthday, so the probability that it is Adam and Eve who have the same birthday is at least $\frac1{10}$, much more than the $\frac1{365}$ it would be without the extra information.
To get the exact value, use $$P(X\mid Y)=\frac{P(X\text{ and }Y)}{P(Y)}.$$ Here $X$ is Adam and Eve having the same birthday, and $Y$ is some two people having the same birthday, so $P(X\text{ and }Y)=P(X)=\frac1{365}$.
Now you just need to find $P(Y)$. It is easier to calculate the probability that no two people have the same birthday, and subtract from $1$. (Hint: what is the probability that the first two have different birthdays? If they do, what is the probability the third person has a different birthday from both of them?)