Combinations and . . . Non-combinations – Part II

In my previous article, I mentioned that the Fundamental Counting Principle (FCP) can be used to solve most GMAT counting questions.  The FCP says:

If a task can be completed in stages, where one stage can be accomplished in A ways, another stage in B ways, another in C ways . . . etc., then the total number of ways to accomplish the entire task will equal A×B×C×… 

If you’re interested in a more detailed explanation of the Fundamental Counting Principle, you can watch this video.

The great thing about the FCP is that it’s easy to use, and you need not memorize any formulas.  So, whenever I encounter a counting question, I first try to determine whether or not the question can be solved using the FCP.  I do this by asking, “Can I take the required task and break it into individual stages?”  If the answer is yes, I may be able to use the FCP to solve the question.  To see how this works, let’s solve the following question:

A “palindromic integer” is an integer that remains the same when its digits are reversed. So, for example, 43334 and 516615 are both examples of palindromic integers. How many 6-digit palindromic integers are both even and greater than 400,000?

(A) 200

(B) 216

(C) 300

(D) 400

(E) 2,500

Can we take the task of “building” 6-digit palindromic integers and break it into individual stages? Yes!

Let’s say that our 6-digit integer takes the form XYZZYX.  Once we choose values for X, Y and Z, we can be assured that the resulting 6-digit integer is palindromic.

From here, we can define the stages as follows:

Stage 1: Choose a value for X

Stage 2: Choose a value for Y

Stage 3: Choose a value for Z

Once we complete all 3 stages, we will have “built” our 6-digit palindromic integer.  So, at this point, we must determine the number of ways to accomplish each stage.

Stage 1: Choose a value for X:  In how many ways can we choose a value for X?  Since the resulting 6-digit integer must be both even AND greater than 400,000, X must equal 4, 6 or 8.  So, Stage 1 can be accomplished in  3 different ways.

Stage 2: Choose a value for Y:  Since there are no restrictions on the value of Y, we can choose any of the ten digits {0,1,2,3,4,5,6,7,8,9}. So,  Stage 2 can be accomplished in 10 different ways.

Stage 3: Choose a value for Z:  Since there are no restrictions on the value of Z, we can choose any of the ten digits {0,1,2,3,4,5,6,7,8,9}.  So,  Stage 3 can be accomplished in 10 different ways. 

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus create an even, 6-digit palindromic number that’s greater than 400,000) in 3 × 10 × 10 ways (300 ways).  So, the answer to the original question is C.

In my next article, we’ll examine some additional considerations when applying the FCP.