Combinations of $5$ cards out of $52$ that don't include $4$ aces How would I calculate the number of different ways (order doesn't matter) I can take out $5$ cards from a deck of $52$ cards, without ending up with $4$ aces? 
A way would be to say that the number of ways I can end up with $4$ aces is $48$; the $4$ aces + any of the remaining $48$ cards, and then subtract this from the total number of different hands possible ($2.598.960$).... but is there any other way? 
In the above method, I calculate the opposite, and then subtract that from the total, but is there any way I can avoid doing this, and directly calculate the number I want? 
Another example would be how many different ways I can get at least one six after having thrown two dices. Again, I would calculate this by calculating the opposite (no six; $5 \cdot 5 = 25$ ) and subtract from the total ($36 - 25 = 11$).... but how would one calculate that $11$ directly, if one wanted to? 
Thanks. 
 A: Well, you could, for the four-aces thing, calculate the probability of getting no aces, of getting one ace, of getting two, and of getting three, and add them together. As you can guess that gets boring: the full formula is
$$\frac{\binom{4}{0}\cdot\binom{48}{5}+\binom{4}{1}\cdot\binom{48}{4}+\binom{4}{2}\cdot\binom{48}{3}+\binom{4}{3}\cdot\binom{48}{2}}{\binom{52}{5}}$$
For the two dice, you can do the same thing:  The probability of getting exactly one 6 is $\left(\frac{1}{6}\right)^1\left(\frac{1}{6}\right)^1\binom{2}{1} = \frac{1}{6}\cdot\frac{5}{6}\cdot2=\frac{5}{18}$, and the probability of getting two 6s is $\frac{1}{36}$, so those add up to $\frac{11}{36}$.
If you have harder problems, like, say -- "roll ten dice, what's the probability of getting 4 or more 6s?" then you're stuck.  You have to do this accumulation no matter which direction you pick.
A: i.e. 
$$\begin{align} 
\sum_{i=0}^{3}{4\choose i}{48\choose {5-i}}&=\sum_{i=0}^{4}{4\choose i}{48\choose {5-i}}-{4\choose 4}{48\choose 1}\\
&={52\choose 5}-{4\choose 4}{48\choose 1}
\end{align}$$
by Vandermonde's Identity
