What is an Ordering Task?
Ordering tasks are a common type of question on the GMAT. In an ordering task, you are asked to determine the number of ways to order a set of objects. The number of ways to order n different objects is n!, or n factorial.
The best way to conceptualize the ordering task is as filling slots. Here’s an example question with a simple graphical representation:
How many ways can 7 cars be lined up in 7 parking spaces?
___ ___ ___ ___ ___ ___ ___
We can imagine these seven slots as the seven parking spaces. How many different options do we have for filling the first slot? Well, since we haven’t assigned any cars to any slots yet, we have all seven cars available.
___ ___ ___ ___ ___ ___ ___
7
What next? No matter which of the seven cars was assigned to the first slot, we have six cars left.
___ ___ ___ ___ ___ ___ ___
7 6
By now you might see where this is going. If we keep filling in the possible options for each slot, from left to right, we get this:
___ ___ ___ ___ ___ ___ ___
7 6 5 4 3 2 1
The n! Formula for Ordering
By the time we get to the last slot, we have assigned every car but the last one, so we have only one choice. Those who know their math notations may recognize the “descending integers” pattern formed in this scenario:
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1
This is just one example of the standard formula for the ordering task. The number of ways to order n different things (spatially or temporally) is n! (pronounced “n factorial”), the product of n multiplied by each positive integer less than n. If we had 10 cars instead of 7 cars, there would be 10! ways to line up those 10 cars in 10 parking spaces. If we had 16 cars, there would be 16! ways to line up those 16 cars in 16 parking spaces.
Ordering Tasks with Identical Elements
Simple enough, but the ordering task can involve an extra twist: what if some of the elements being ordered are considered identical or indistinguishable? In other words, we count all orders in which these identical elements trade places with each other (while other elements stay put) as identical orders.
The Overcounting Problem
Consider the word INTUITION. How many different ways can the letters in this word be arranged? You might be quick to answer “9 factorial,” since there are 9 letters. However, this would overcount the number of different ways the letters can be arranged, since any arrangements where identical letters trade places are not recognizably different.
Let’s say we arrange the letters like this:
INTIOUNTI
Technically, those three I’s can swap places amongst themselves, and we would never be able to tell. Likewise, the two T’s or the two N’s can trade places with each other, and we would never be able to tell. This is true for any arrangement of the letters – not just in our example. But the “n factorial” method counts each one of these identical arrangements separately, thus significantly overcounting the number of different arrangements.
The Solution
The good news is that we don’t have to start from scratch: we can correct this overcount using our existing understanding of ordering. Let’s consider each group of identical letters as its own isolated group. Since there are three I’s, the number of ways to order the I’s independently is 3!
So for any given arrangement of the letters in the word INTUITION, there are 3! ways that the I’s can swap places amongst themselves without us noticing. Likewise, and simultaneously, there are 2! ways for the N’s to trade places and 2! ways for the T’s to trade places.
The “n factorial” counting method for ordering will count each of these indistinguishable arrangements separately. So we must divide the “n factorial” number by the number of indistinguishable arrangements hiding within each different arrangement. Since the word INTUITION has 9 letters, 3 I’s, 2 N’s, and 2 T’s, the number of different arrangements for the letters is 9! / (3!*2!*2!).
When identical or indistinguishable elements are present, we divide the factorial of the full number of elements (n factorial) by the product of the factorials of each group of identical elements.
Conclusion
In this article, we have introduced the concept of ordering tasks and discussed how to handle them when the elements being ordered are identical. In the next article, we will explain permutations on the GMAT.
If you are planning to take the GMAT, and need guidance through your journey, schedule a free consultation call with a top-scoring tutor to get your best score. Get a customized study plan and receive one-on-one instruction.
Contributor: Elijah Mize (Apex Instructor)