The title of this post may seem facetious. After all, even the person most allergic to math, most traumatized by math, still remembers how to count! The QUANT, of course, generally will not ask you, for example, to count from one to seven. The QUANT may give you a more complex scenario, and ask you to count how many ways can such-and-such happen. For example
1) Shakespeare wrote fifteen comedies (including the so-called "romances"), ten histories, and twelve tragedies. If a summer Shakespeare festival always has one comedy, one history, and two tragedies, how many different combinations of plays can the festival host?
As you see, this "counting" is a little more challenging than the kind of "counting" you learned in your salad days. I would like to convince you, though, that you are quite capable of solving problems like this.
This one big idea will give you a lot of mileage on any of the problems where the QUANT asks you to count things.
If option #1 has
The FCP easily extends from two choices to three or any higher number. However many collections of alternatives there are, you simply multiple the number of alternatives in each set to produce the total number of combinations. For example: Shakespeare wrote fifteen comedies, ten histories, and twelve tragedies. If we are going to pick one of each kind, and ask how many different trios of plays can we create, the total number is simply
Before we can answer the original question posed, we have to clarify some terminology about counting. A permutation is a set in which order matters —
This distinction is important in counting because we have to know whether to include the sets that repeat elements in different order. In question #1 above, the question explicitly asks for combinations. In other words, if we pick Hamlet and then King Lear, that’s will be considered the same as picking King Lear and then Hamlet.
This Web Page considers permutations and combinations in greater detail. For the purposes of this post, we just have to be careful to consider what we are counting.
Shakespeare wrote twelve tragedies, and we want to pick a pair from these. How many different pairs can we pick? We will use the FCP.
Let’s break the task into a first choice and a second choice. For the first choice, we can choose any of the
If we were interested in permutations, then
Now, we can answer the entire question. We want the number of combinations of four plays consisting of one comedy, one history, and two tragedies. By the FCP, that’s
Switch the order —
Use doubling-halving on the
Then, the easiest, multiplying by ten —
Thus, the festival can come up with
A certain restaurant offers
To count the number of meals, we have to count the possibilities for the three components, and then, according to the Fundamental Counting Principle, we multiply.
For salads, there are
For main courses, there are
For desserts, there are
So, there are
For the number of meals, we multiply
Answer = (D)