Problem Dissection: GMAT Combinations and Permutations Part 1
Welcome back. As I mentioned before, each week we are going to feature a problem breakdown here on the blog. Last week we dove into probability. We’ll revisit that topic in the weeks to come. This week we are going to start moving into combinations and permutations. These can be some of the toughest problems on the test. However, since I can feel your pain on this topic, we are going to start slow. Instead of diving in, we’ll start by getting our big toe wet first, or maybe our little toe. For those who are at an advanced level on this topic, check back in a few weeks. We’ll be up to speed and breaking down some tough GMAT problems. For now, let’s use a problem to get a feel for how we handle these things.
Kim has four trophies, which she wishes to display in a cabinet with five shelves, with only one trophy to a shelf. How many different ways are there to arrange the trophies?
This is an arrangement question. Order matters here (you should always be asking yourself if order matters). Often you can simply count the different possible arrangements, making sure to do so systematically, but it is usually easier to remember that the number of ways of arranging x objects is x !
(Note: Remember, the symbol, ‘!’ is a factorial function, to multiply a series of descending natural numbers.)
There are too many arrangements here to count by simply writing them out, so you will have to use factorials to count the arrangements.
In this case you have 4 objects, but 5 spaces. So you have five possibilities for each shelf. It could contain one of the four trophies, or it could be empty. So the number of possible arrangements is 5! (5x4x3x2x1) = 120. The answer is (E)