Lecture 3 - Ch 4 - Counting Rules

These counting rules are basics from the GMAT, so it's mostly review. Getting these problems right is a matter of understanding the scenario and identifying how to apply the two simple formulas for permutations and combinations.

All of these cases involve selecting some number, call it x, of items from a pool of n possible items.

Permutations: Use this formula when order matters in the scenario. I.e. if (a,b) is not the same as (b,a). An example of this would be license plates. The license plate "GO4MBA" is not the same as "BAG4MO". Another example is telephone numbers. Obviously, 867-5309 is not the same number as 509-3678.
Combinations: Use this formula when order doesn't matter in the scenario. I.e. if (a,b) is considered the same as (b,a). In this case you start with the same formula above, but you need to divide by the number of possible duplicates (x!) due to the fact that order doesn't matter. The formula now becomes:
When dealing with questions of probability, we often need to use these formulas to figure out the number of successful events and the total number of possible events. For example, the probability of being dealt a royal flush from 5 cards would be the total number of ways to get a royal flush divided by the total number of possible hands that could be dealt.

There are more complicated examples involving replacement and other factors, but for our purposes, I'll leave it at this and perhaps bring those up in a side note blog entry.

For those interested in TeX/LaTeX using LEd and MiKTeX 2.7, apparently they don't support the \binom control sequence. If it did, I could just create the symbol above with:
\binom{n}{x}
Since it doesn't, the best I could do is to use an array and code it with:
\left(\begin{array}{c}n\\x\end{array}\right)
Yuck!