#5 - The permutation formula and other unnecessary formulas to forget

Question

https://i.imgur.com/RW5584j.png

Each year, the GMAT test-makers publish the GMAT  Official Guide  (the OG), which includes all of the properties, formulas, and techniques tested on the GMAT. I call these the   core tools  . 
In the OG2022, these core tools are summarized on pages 94 to 106, and they include: 
 • The “or” probability formula: P(E or F) = P(E) + P(F) – P(E and F)
• This remainder property: If y divided by x equals q with remainder r, then y = xq + r
• The quadratic formula: If ax2 + bx + c = 0, then x = -b ± √(b2 . . . etc.

Important: All official GMAT quant questions can be solved by applying one or more of the core tools noted in the OG. In other words, the test-makers will never create a math question that requires tools beyond those presented in the Official Guide.  So, if you can learn how to apply those versatile core tools in a variety of circumstances, then you’ll be prepared to handle anything the GMAT can throw at you.
Nevertheless, most GMAT students will encounter a wide range of additional, non-core formulas during their studies. Despite their potential allure, most of those formulas aren’t worth your time. 
Don’t get me wrong. I totally understand why a lot of students embrace formulas of all kinds. For many, they appear to provide a sense of structure in a topic that often seems like no more than a random collection of rules and properties.

Unfortunately, memorizing a lot of non-essential (i.e., non-core) formulas can actually make you less effective at solving GMAT math questions and should be avoided for several reasons: 
 1) Many provide no value at all. 
2) Many are too specific and don’t prepare you for alterations in a question. 
3) Learning and practicing non-essential formulas increases your prep time and prevents you from mastering the core tools the test-makers EXPECT you to master. 
4) The test-makers are unlikely to create new questions that reward students for memorizing non-essential formulas

1. Many provide no value at all 
When it comes to the topic of counting, the Official Guide provides 3 core tools: 
 • The Fundamental Counting Principle (aka FCP, aka multiplication principle, aka slot method)
• n distinct objects can be arranged in n! ways (an extension of the FCP)
• The combination formula

The FCP alone can be used to solve tons of counting questions, including this one I made up:
How many 2-letter codes can be created from the letters A, B, C, D, E and F such that no code contains repeated letters?
(A) 12
(B) 24
(C) 30
(D) 36
(E) 72

This question is perfectly suited for the FCP. 
There are  6  options for the first letter and, to avoid duplication, there are  5  options remaining for the second letter. 
By the FCP, the total number of ways to select the first and second letters (and thus create a 2-letter code) = ( 6 )( 5 ) =  30  =  C

Many readers will note that this question can also be solved via the permutation formula, which is really just a special case of the FCP. 
The permutation formula goes something like this:
 If nPr represents the total number of ways to arrange r objects from n unique objects, then nPr = n!/(n-r)! 
For my question above, we’re arranging 2 letters from 6 letters. 
So, the number of 2-letter codes possible = 6P2 = 6!/(6-2)! = 6!/4! = 30

Since both solutions take about the same amount of time, the permutation formula doesn’t provide any advantage. In fact, there aren’t  any  official GMAT questions where the permutation formula provides an advantage. Given this, there’s no reason to memorize it.

Note : The pitfalls of memorizing non-essential formulas are reduced (but not eliminated) if you understand  how  and  why  those formulas work. For example, if you understand how and why the permutation formula works, then you may be able to apply that rationale to other questions.

The paradox with the permutation formula is that, if you understand how/why it works, then you’ll also understand why it’s unnecessary.

2. Many non-essential formulas are too specific and don’t prepare you for alterations in a question  
Let’s examine another official counting question.

Note  : I believe this next question is from the old official PowerPrep practice tests and isn’t on any of the current official practice tests. I could be wrong. So, continue at the potential risk of seeing a question from a practice exam you may take in the future.

At a dinner party, 5 people are to be seated around a circular table. Two seating arrangements are considered different only when the positions of the people are different relative to each other. What is the total number of different possible seating arrangements for the group?
(A) 5
(B) 10
(C) 24
(D) 32
(E) 120

Shortly after this question made its first public appearance on the GMAT forums (circa 2005), a new formula emerged that would make short work of this new question “type.”

The formula is an extension of the FCP, and it goes something like this: 
 n objects can be arranged in a circle in (n–1)! ways (if arrangements are considered different only if the objects’ relative positions are different)

Applying the formula to the given question, we see that the 5 people can be arranged in (5-1)! ways. So, the correct answer = 4! = (4)(3)(2)(1) = 24 = C. 
Although it quickly led us to the right answer, the circular arrangement formula is a crutch that actually makes us less able to solve seemingly-similar questions.

For example, consider this question I created for GMAT Club: 
 https://i.imgur.com/NTdaSDC.png 
At a dinner party, 6 people (A, B, C, D, E, and F) are to be seated around the circular table shown above. Two seating arrangements are considered different only when the positions of the people are different relative to each other. If person B must sit directly across from person C (e.g., B in chair #1 and C in chair #4), what is the total number of different possible seating arrangements for the group?
(A) 20
(B) 24
(C) 48
(D) 72
(E) 144

As you can see, I made two minor changes to the original question: The number of people is increased from 5 to 6, and B must sit directly across from C. 
Nevertheless, the success rate for this modified question is only 53%, whereas the success rate for the original/official question is 74%.

Is the modified question that much harder than the original question? 
It depends. 
 • If you understand how and why the circular arrangement formula works, then you may be able to apply some quantitative reasoning to the circular arrangement formula and solve the question, just as Vaibhav did  here . 
• If you don’t understand the rationale behind the circular arrangement formula, then you’ll have a hard time applying it to the new question. 
• If you already know how to apply the FCP to questions involving circular arrangements, then the two changes to the question have little impact on the difficulty level of the altered question. In fact, you’ll see that this  solution  to the original question is almost identical to this  solution  to the altered question. Why? Because the same core tools were used in both solutions.

Are all non-essential formulas bad?  
No. There are some non-core formulas worth knowing, but if you’re going to add any non-core formula to your mathematical toolbox, it should (most times) meet two criteria: 
 • It’s flexible enough to be applied to more than one super-specific situation
• You have a basic understanding of why/how it works, so you can make any necessary adjustments if a question doesn’t quite match up with the formula

I believe the Mississippi rule is one of the non-core formulas worth learning, because it’s a minor extension of the FCP that we can use to solve questions like these  easy ,  medium  and  hard  questions from the Official Guide.

It’s worth noting that the decision to add the Mississippi rule to my course wasn’t easy. In fact, on any given day, I can be convinced that adding it was unnecessary. Also, here’s  how/why the formula works .

Another example.  
You may be surprised to learn that the Official Guide lists just one core tool related to speed-distance-time questions:  distance = (rate)(time) .

This one formula can be applied to a wide variety of questions, including this one from the Official Guide: 
A car travels from Mayville to Rome at an average speed of 30 miles per hour and returns immediately along the same route at an average speed of 40 miles per hour. Of the following, which is closest to the average speed, in miles per hour, for the round trip?
A) 32.0
B) 33.0
C) 34.3
D) 35.5
E) 36.5

Although the core  distance = (rate)(time)  formula can be used to solve this question (as well as all other official speed-distance-time questions), many students will memorize non-essential formulas such as this: 
 If an object travels from A to B at a speed of x and then immediately returns to A at a speed of y, the average speed for the round trip = 2xy/(x + y)

When we apply this formula to the Mayville-to-Rome question, we let x = 30 and y = 40, which means the average speed for the round trip = 2xy/(x + y) = 2(30)(40)/(30 + 40) = 2400/70 = 240/7 ≈ 34.3. Answer: C. 
The non-core formula works perfectly here, but, if you check these  two solutions , you’ll see that the test-makers’ core tool works too. 
 
 If both formulas work, what’s the issue?  
One issue is that the average-round-trip formula above applies only to situations in which something travels from A to B at one speed, and then travels from B to A at a different speed.

The bigger issue is that, if you don’t understand how/why the formula works, you’ll likely have difficulties answering similar, yet slightly altered, questions. 
For example, what if the car stopped in Rome for 15 minutes before returning to Mayville? Do we need to adjust the formula? If so, how?

What if, upon returning to Mayville, the car turned around and drove back to Rome at 30 miles per hour? What would be the average speed for this three-part trip? Can the formula be modified to handle this alteration?

As you can see, the non-essential formula falls apart for those slightly altered questions, whereas the core distance = (rate)(time) formula is flexible enough to handle every alteration.

There are many examples of super-specific formulas that fall apart when they encounter the slightest alterations, whereas the core tools provided in the Official Guide are so versatile they can be applied to any GMAT question.

Here’s the worst aspect of non-essential formulas: 
 3. Learning and practicing non-essential formulas increases your prep time and prevents you from mastering the core tools the test-makers EXPECT you to master.  
Let’s say, while studying distance-rate-time questions, you decide to memorize and practice 9 non-essential formulas as well as the 1 core formula. 
Some might feel that knowing 10 formulas is 10 times better than knowing 1 formula, but this isn’t the case at all, because learning all of those formulas causes problems. 
First, your prep time will be increased if you must learn and practice 10 tools instead of 1. Second, when you encounter a distance-rate-time question on test day, you must now correctly determine which of the 10 formulas to apply, and, even if you do choose the correct formula, will you have a deep enough understanding of it to make the necessary modifications if the question doesn’t quite match the parameters of the formula?
If not, you’ll be forced to apply the core distance-rate-time formula, but if you haven’t mastered the many different ways to apply that flexible core tool, you may get the question wrong.

Another example  
Try answering this question from the Official Guide before continuing.

Jack is now 14 years older than Bill. If in 10 years Jack will be twice as old as Bill, how old will Jack be in 5 years? 
(A) 9
(B) 19
(C) 21
(D) 23
(E) 33

Some students will solve this question algebraically, others by testing the answer choices, and others by applying this non-essential formula I just created: 
 If A is presently x years older than B, and A will be k times as old as B in y years, then A’s age in q years = (yk + xk – y)/(1-k) + q

When we apply this formula to the question, we see that x = 14, k = 2, y = 10 and q =5 
So, Jack’s age in 5 years =  /(1 – 2) + 5
=  /(-1) + 5
= (–18)/(-1) + 5
= 23
Answer: D

The formula works, but is it worth memorizing? 
Definitely not. 
I could create a separate formula for every official quant question ever posed, but even if you spent thousands of hours memorizing and practicing them, you’d still have difficulties on test day, because . . .

4. The test-makers are unlikely to create new questions that reward students for memorizing non-essential formulas 
This excerpt from the Official Guide details the test-makers’ position: 
 Knowledge of basic math, while necessary, is seldom sufficient for answering GMAT questions. Unlike traditional math problems you may have encountered in school, GMAT Quantitative Reasoning questions require you to  apply  your knowledge of math. For example, rather than asking you to demonstrate your knowledge of prime factorization by listing a number’s prime factors, a GMAT question may require you to  apply  your knowledge of prime factorization and exponents to simplify an algebraic expression with a radical.

The test-makers are true to their word. Rather than ask us to determine the prime factorization of some number, they require us to apply our knowledge of prime factorization to answer questions from various Official Guides such as  this  and  this  and  this  and  this  and  this  and  this  and . . . you get the idea.

On the other hand, among the thousands of former exam questions the test-makers have published in the Official Guides, not one can be directly solved via the non-essential circular arrangement formula. It’s also unlikely they’ll create another standard version of this question again, because the test-makers don’t create questions that reward students for simply memorizing non-core formulas, especially if students don’t understand how/why those formulas work. 
Instead, the test-makers will continue creating questions that test our quantitative  reasoning  skills.

Final Words  
Many non-essential formulas aren’t flexible enough to provide any value beyond solving the one question the formula was based on. So, rather than load up on extra formulas, get really good at applying the core tools the test-makers expect you to master.

I want to thank  Ian Stewart , GMAT mathematician extraordinaire, for reviewing this article. Any remaining errors are my own. 
 Note from Ian: I agree completely with Brent’s central point — if you’re studying for the GMAT by memorizing a lot of formulas, you’re not learning what the GMAT actually tests — but I’d go even further. Even some of the formulas presented in the Official Guide Math Review are unnecessary on the GMAT.

vv65 · Answer

IanStewart  , I forgot the quadratic formula long ago and have never needed to use it for any GMAT question. But then, I don't spend enough time on quant reasoning.

Are there official questions that need the quadratic formula? Would you recommend that test-takers learn it?

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IanStewart · Answer

That's actually why I asked Brent to include my note at the end of his article -- I've never once seen an official question where you'd want to know the quadratic formula, assuming you know how to factor a standard quadratic. As Brent points out, the quadratic formula does appear in the OG math review section, so I suppose in theory it could be tested, but I've never seen an official question where it would even be helpful. Official questions don't test whether you've simply memorized a formula and can plug numbers into it, so it's hard to imagine what an official question testing the quadratic formula would even look like.

That said, there are rare questions where you could, as an option, use one detail from the quadratic formula to solve. If you look at the quadratic formula, under the root we see the expression b^2 - 4ac, which is sometimes called the 'discriminant' of the quadratic ax^2 + bx + c. When the discriminant is positive, the quadratic equation ax^2 + bx + c = 0 will have two different solutions for x, but when the discriminant is zero, this equation has only one solution (and it has no solutions if the discriminant is negative). So you can sometimes use the discriminant in GMAT DS questions to decide if a quadratic will give you one solution or more than one, which is what we care about in DS. As I said, though, using the discriminant is optional; if you know how to factor a quadratic, there will be simpler alternatives you can use.

vv65 · Answer

Thank you! That confirms my impression.

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#5 - The permutation formula and other unnecessary formulas to forget

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