Probability that exactly k of N people matched their hats [SRoss P63 Ex 2g] 
The match problem stated in Example 5m of Chapter 2 (of A First Course in Pr, 8th Ed, Ross) showed that the probability of no matches when $N$ people randomly select from among their own $N$ hats $= P[N]= \sum_{0 \le i \le N}(-1)^i/i!$
  What is the probability that exactly $k$ of the $N$ people have matches?
Solution: Let us fix our attention on a particular set of $k$ people and determine the
  probability that these $k$ individuals have matches and no one else does. Letting $E$
  denote the event that everyone in this set has a match, and letting $G$ be the event that
  none of the other $N − k$ people have a match, we have
  $P(E \cap G) = P(E)P(G|E)$ 
  (Rest of solution pretermitted)
$P(E) = \dfrac{\text{ 1 choice for C1 } \times ... \times \text{ 1 choice for C(k - 1) } \times \text{ 1 choice for C(k) } \times N - k \text{ choices for C(N - k) }\times \ N - k - 1 \text{ choices for C(N - k - 1)} \times ...}{N!}$
  , where $C(k) =$ chap $k$, chaps 1 through k each has one choice due to their success in finding their hat, and the $P(E \cap G) = P(E)\binom{N}{k}P[N - k]$. 

I see that $P(E) \neq P(E \cap G)$, but I don't apprehend the method and still deem $G$ redundant.
Since $E$ is the event that exactly these $k$ people, for some $k$, have a match,
how and why isn't the required probability just $P(E)$? 
Since there are only $N$ people, thus the occurrence of $E$ (coincidently, directly, and straightaway) equals the occurrence of $G$?    

Okay. Let us think about it step by step.
Step 1: What do we want? We want that EXACTLY k people should have matches. The rest, i.e the remaining N-k people should NOT have matches. So let A = the event that exactly k people have matches and B = the event that N-k people have matches. Then, we want to know P(A$\bigcap B$).
Now, notice that from the Bayes formula, we can calculate this probability by either P(A|B)P(B) or P(B|A)P(A). Note also that the first is much harder to calculate. So we want to calculate the probability that N-k don't have matches given that k people have matches.
So let's do that! The probability that P(B|A)P(A). Note that we know P(B|A) like you mentioned from doing the earlier problem, because this is just the probability that N-k people don't have a match which should just be $P_{N-k} = \sum(-1)^{i}/i!$ for i = 0 to i = N-k.
And all you need to calculate is the probability of exactly k people getting a match which is $P_{k} = \frac{1\times 1\times 1.......\times1}{N\times N-1 \times N-2.....1}$, which is basically saying that the first man picks his hat, times the second man picks his hat, times the third man picks his hat...... Hope this helps!
