The Mississippi Rule Explained

By Brent Hanneson - April 16, 2022

Each year, the GMAT test-makers publish the GMAT Official Guide (aka the OG), which lists all of the properties, formulas, and techniques tested on the GMAT. 

When it comes to the topic of counting, the OG provides us with 3 basic tools: 

• The Fundamental Counting Principle (FCP, aka slot method, aka multiplication principle)
• n distinct objects can be arranged in n! ways
• The Combination formula

Although all official counting questions can be solved by applying one or more of those 3 basic tools, it’s sometimes useful to add extra tools to your mathematical toolbox. However, if you decide to add any new tools to your toolbox, try to understand how and why those tools work. 

In this brief article, we’ll examine how and why the Mississippi Rule works.  

When we want to arrange a group of objects in which some of the objects are identical, we can apply the Mississippi Rule, which goes something like this:

Mississippi Rule - Formula.png

To show why this formula works, let’s calculate the number of ways to arrange the letters in the word NANNY, in which we have 3 identical N’s.

We’ll start by making the 3 N’s different by labeling them N₁, N₂ and N₃. 

Now let’s arrange the objects N₁, N₂, N₃, A, and Y. 

Since we now have 5 distinct objects, we can arrange them in 5! ways (i.e., 120 ways) as per one of the given properties provided in the OG. 

One of those 120 arrangements is N₁N₂AN₃Y, in which the three N’s are in the first, second and fourth positions. 

Notice that we can arrange the 3 N's in these same 3 positions in 6 ways as follows: N₁N₂AN₃Y,   N₁N₃AN₂Y,   N₂N₁AN₃Y,   N₂N₃AN₁Y,   N₃N₁AN₂Y,   N₃N₂AN₁Y.

Since the three N’s are actually identical, all 6 arrangements represent the single word NNANY. 

Similarly, the following 6 arrangements all represent the single word YANNN:   YAN₁N₂N₃,   YAN₁N₃N₂,   YAN₂N₁N₃,   YAN₂N₃N₁,   YAN₃N₁N₂,   YAN₃N₂N₁

Key point: Since the 3 letters, N₁, N₂ and N₃, can be arranged in 3! ways, every unique arrangement of the letters in NANNY is counted 3! times (i.e., 6 times)

In other words, when we counted 120 (aka 5!) arrangements, we actually counted each unique arrangement 6 times (aka 3! times). So, to compensate for this duplication, we must divide 5! by 3!, which means we can arrange the letters in the word NANNY in 5!/3! ways. 

We can follow a similar process to arrange the letters in the word NNNNAY. If we first label the 4 N’s as N₁, N₂, N₃ and N₄, then we can arrange the 6 unique objects (N₁, N₂, N₃, N₄, A and Y) in 6! ways. 

However, for each possible arrangement, we can arrange the 4 N’s in 4! ways yet still have the same word. So, to compensate for this duplication, we must divide 6! by 4!, which means there are 6!/4! ways to arrange the letters in the word NNNNAY.   

So, to count the number of ways to arrange the letters in the word ABBCCCC, we’ll take 7! (the number of ways to arrange 7 unique objects) and divide it by 2! (the number of ways to arrange the 2 B’s if we were to label them B₁ and B₂) and also divide it by 4! (the number of ways to arrange the 4 C’s if we were to label them C₁, C₂, C₃ and C₄). 

So, the letters in the word ABBCCCC can be arranged in 7!/2!4! ways (105 ways).  

And that’s it! 

If you’re interested, here are two official GMAT practice questions where we can apply the Mississippi Rule: 

• 500-level question | My solution
• 700-level question | My solution