Are you intimidated by Permutations and Combinations questions in GRE Math?

Do you scratch your head every time you are solving a question on P&C?

Well, you aren’t the only one! Don’t worry, most students who prepare for the GRE are stumped by questions on Permutations and Combinations. However, P&C is an integral chapter of GRE Math.

Gaining your mastery in this topic is a decisive aspect if you wish to achieve that dream score of 170!

Download our eBOOK on “Quant handbook for a perfect 170 score“

Why Is Permutation and Combination Important in GRE Math?

GRE Math measures your competency in four primary areas: arithmetic, algebra, geometry, and data analysis. Permutation and Combination��is one concept falling under the data analysis section. But P&C is just a small part of data analysis – you will typically get only a few questions on it.

Now, we know what you are wondering: ‘Why must I focus on Permutation and Combination if it forms such an insignificant part of my GRE Math?’

Well, the answer is simple: It’s to differentiate the great from the good! If you are aiming for a high score in the Quantitative section, and have already mastered the rest of the topics such as algebra, arithmetic, and geometry; then a few questions can make a world of difference to your scores. And not just for your scores, the same few questions could make a difference between an admit into an elite institute as opposed to an admit into a mid-tier university.

It is therefore important that you familiarize yourself with the basics of Permutation and Combination, thereby raising your chances for a higher and a better GRE Quant score. And, we are here to help you do exactly that!

So, let’s begin with the basics. Permutations and Combinations are the bane of many students. But do you know the difference between permutation and combination? No? Don’t worry, you’re not alone.

What Are Permutations and Combinations?

As mentioned earlier, most students aren’t aware of the difference between the two concepts. While the difference is small, being aware of it will lead to accurately applying the correct formula at the correct junction.

Permutations: Let’s begin with Permutations. One way to think of this concept is as the number of arrangements or orderings within a fixed group.

For instance, if I have five students and wish to figure out in how many ways they can sit on five different chairs, I will have to use the permutations formula. This is because the number in the group is fixed. Secondly, we have to examine the number of ways in which we can “arrange” the students on the five chairs.

Combinations: On the other hand, the combination formula is useful when figuring out how many groups can be formed from a larger pool of people. For example, if a teacher wanted to know how many distinct groups she can form from all the people in the class, she would use combinations. The imperative word here is “distinct,” as one student can’t be in two groups at once.

To simplify:

Permutations & Combinations – GRE Math

Here is one worked out example of each kind to help you understand better.

1) In how many different ways can we award the 1st, 2nd, and 3rd place prizes among eight contestants? (Gold / Silver / Bronze)

In this question, we have to arrange 8 people in a sequence. We will use the permutations formula here since the order is important in solving this problem.

Here is how it would work out:

• For the first place spot, we have 8 people to choose from: A B C D E F G H
• As we go down the list, the number of people we have to choose from will decrease by one each time.
• Therefore, silver medal will have 7 choices: B C D E F G H. [B wins the silver]
• Bronze medal – 6 choices: C D E F G H. [C wins the bronze]
• Here you see that we picked specific people to win. We had 8 choices at first, then 7, then 6. The total number of options was 8 × 7 × 6 = 336
• But we don’t have to do 8, since we have to order only 3 out of 8 people.

Because 5 people were left after we picked 3 medals, the final formula is:

�� �� �� �� �� �� �� ����[which translates to]�� �� �� �� �� �� ��

Now, consider this question:

2) Steven draws 13 portraits for his art project. He wants to choose 8 of them for his studio art final. How many different groups of photos are possible?

If you observe carefully, this question asks about ‘groups’ and not the order in which the portraits will be chosen. Therefore, using the formula,

=��13!8! 13-8!

= 1287 different groups

Pro Tip: Always remember to ask yourself whether the order matters in the question. Be mindful while solving P&C questions, as the GRE may also combine one or more of these concepts into the same problem!

• Addition Rule: If an experiment can be performed in ‘n’ ways, and another experiment can be performed in ‘m’ ways, then either of the two experiments can be performed in (m + n) ways.
• Multiplication Rule: If an operation can be done in ‘m’ ways, while another operation can be done in ‘n’ ways, then both of the operations can be performed in m × n ways.

Take a GRE style practice quiz on Permutations & Combinations now!

��Tips to Master Permutation and Combination Questions in GRE Math

To be able to ace your GRE math, you have to be at the top of your game all the time! While this may seem daunting at first, with the right techniques and appropriate guidance, you will be surprised at how easily you can achieve a perfect score of 170 in Quant. But blind hard work isn’t enough; in fact, with an exam such as the GRE, working smarter rather than harder is the key!

To help you reach your dream score, we have compiled a list of some of the best tips and tricks you can use to ace your Quant section, especially when it comes to the dreaded P&C! Here they go…

1) Confirm That the Question Is Actually on P&C

First and foremost, it is imperative to understand whether the question itself requires you to use the concepts of Permutations and Combinations. In GRE Math, questions don’t come according to chapters or labels as is the case with your practice books. Rather, they are a concoction of different topics and concepts thrown together in a blender mix of a section! You can have a P&C question sandwiched between an arithmetic and geometry question. It is therefore important to determine whether the problem involves permutations and combinations or not, and not keep a one-dimensional mind.

One simple way to be alert to this is to, of course, look out for the word ‘����Dz������������ٲ�’ in the question statement. However, they usually won’t be so straightforward. Thus, if you cannot find the word ‘����Dz������������ٲ�’ embedded in the question, then keep an eye out for other important markers and related keywords. All of these imply the use of Permutations and Combinations.

Some of these could be:

• Event
• Outcome
• Random Selection
• Chance

2) Memorize the Formulas

The key to acing your Quant section – not just Permutations and Combinations – is memorizing your formulas. Not only is it a healthy habit, but also saves up on crucial time (remember, you only get 2 minutes per question – you can’t waste any precious seconds on recalling formulas!

According to our data, most of our top scorers actually spend more time learning the formulas and concepts rather than practicing the sums! We’ve listed some of the important formulas you can learn and keep in handy:

• The probability of an event = Favorable Number of Events Number of Total Outcomes
• Odds in favor of an event = Number of Favorable Outcomes Number of Total Outcomes
• Odds against an event = Number of Unfavorable outcomes Number of Total Outcomes

3) Pay Attention to the Language

GRE Math in your exam is as much a measure of your analytical abilities as it is about your cognitive prowess. To ace this exam, you have to always be on your toes. The test-makers don’t only want to see whether you can robotically apply formulas; rather they want to investigate whether you can comprehend what is asked of you and suitably apply your brains.

Thus, they create challenging problems by using subtle language to indicate whether you should use a combination formula, a permutation formula, or perhaps both sometimes, to answer the question on hand.

Keep in mind, combination-based questions will indicate that you need to form groups or sets, whereas permutation questions will hint towards finding an order. Words and phrases such as ‘first’, ‘second’, or ‘third’; or ‘how many different ways’ are usually indicators of permutations.

4) Practice, Practice, and more Practice!

One should always pay heed to old adages, for they hold the most wisdom. And it doesn’t get much better that, “practice makes perfect”. There is a reason this has reverberated through history; it’s because it’s true. There is no shortcut to success. The more of these problems you solve, the more you will familiarize yourself with them. By practicing more, even if you get them wrong, you will be acquainting yourself with the style of words, the diction, the way the questions are posed, and the format in which they will be presented to you.

Our data reveals that more often than not, the difficulty bar of questions on P&C is usually set at medium to high. So, make sure that you practice GRE questions which are a little tough. You must practice advanced GRE questions of level 4 and level 5 to be able to tackle the question on your D-Day head on!

Moreover, you will also be reading the corresponding explanations along with the answers, which will further help you understand the concepts of P&C better. The better prepared you are to ace Permutation and Combination questions, the higher your chances of acing your GRE Math!

Special Pointers to Follow On the D-Day

We have also put together a list of some special pointers for you to follow right before you Big Day! Consider this as a practical ‘last minute’ tips list. Read it right before you walk into your exam – and walk out with flying colors!

• Firstly, and most importantly, sleep. And sleep well. Nothing ruins your chances of getting a good score as much as a tired mind. Research has repeatedly shown the importance of a good night’s sleep before and exam. Not only does it refresh you and recharge your batteries, psychologists also say that it helps consolidate and retain memorized information better.��
• Stay calm, composed and as the famous book Hitchhiker’s Guide to the Galaxy states: Don’t Panic! Nothing shuts down your neuro-receptors faster than anxiety. If you feel stuck or are unable to solve a question, move on. Circle back to it later.
• In relation to the above point, if you still feel like you don’t know the answer to the question; then as a last resort make an educated guess (And hope Lady Luck smiles on you). After all, you miss 100% of the chances you don’t take. ����
• Go fully prepared with your stationary in place. Dress well, and eat something filling. Feeling prepared and ready is crucial! Researchers have regularly found correlations between mentally ‘feeling ready’ and associated better performance on cognitive tasks.
• Punctuality is key! The GRE is stressful as it is, so do yourself a favor and reach your test center on time. Worrying about being tardy will only disrupt your mental peace.
• When it comes to P&C, remember:

a) Re-ordering letters of a word, tossing of coins, rolling dice, and pack of cards all come under permutations.

b) Combinations are when you are asked to compute probabilities for independent events that occur together, and mutually-exclusive events.

We hope this article provided you with smart ways to crack Permutation and Combination questions in GRE Math. If you wish to take a GRE style practice quiz on this topic, you can do so here.

All the best!

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Introduction to GRE Probability

The Quants section of GRE assesses you across four broad areas- arithmetic, algebra, geometry, and data analysis. Probability is one of the concepts tested in the data analysis section. However, Probability is just a small part of data analysis and you may expect hardly two to three questions on probability.

Now, you might be asking why to study probability if it forms such an insignificant part of GRE Quants? Well, if you are aiming for a high score in the quantitative section and have already mastered the algebra, arithmetic, and geometry; then two or three questions can make a world of difference in your scores. It is therefore important that you familiarize yourself with the basics of probability thereby raising your chances for a higher and a better GRE Quants score.

Download our eBOOK on “5 Techniques to score a perfect 170 in GRE Quant“

Probability questions can take on many different forms such as multiple choice questions, quantitative comparison, and numeric entry. In case of multiple choice questions, you will be asked to select only one answer from a list of five choices. Quantitative comparison questions consist of two quantities, let’s say A and B, which are accompanied by a description. Your task is to compare the quantities and select one of the four possible choices. Numeric entry questions, as the name suggests, ask you to enter your answer either as an integer or as a decimal in an answer box or enter a fraction in two separate boxes.

Concepts tested in GRE Probability

Now that you have a fair idea about the number and kind of questions to expect, let us have a look at the concepts tested by these probability questions. All GRE probability questions will test your knowledge of two basic concepts- simple probability and permutation and combination.

Simple probability:

Probability is a way to describe uncertainty and the possible outcomes of an experiment using numbers. Simple probability determines the likelihood that an event will happen. Let us look at an example to understand simple probability better.

When we perform an experiment, then the set of all the possible outcomes is called the sample space (S). When we roll a die, the possible outcomes are 1, 2, 3, 4, 5, and 6. Here, in our example of rolling a die the sample space is-

S = {1, 2, 3, 4, 5, 6}

Any subset of the sample space is called an event. So, let’s say “rolling an even number” is an event which corresponds to the outcome of {2, 4, 6}. Now, the probability of rolling an even number is the number of outcomes that constitute the event divided by the total number of possible outcomes. We call the outcomes in an event its ‘favourable outcomes’. So, the probability of an event is determined using the following formula-

Probability = Favourable number of events / total number of events

or, ����������P(E) =n(E) /��n(S)

Therefore, the probability of getting an even number when we roll a die is-

Probability of getting an even number = {2,4,6} /��{1,2,3,4,5,6}

= 3/6 ����

= 1/2 ����

Thus, the probability of getting an even number while rolling a die is half. Or in other words, the probability of getting an even number when we roll a die is fifty percent.

Permutation and combination:

Apart from simple probability, the GRE probability section also tests your knowledge of permutation and combinations. In very simple words, permutation refers to sequences and combination refers to groups. In a sequence order is important. Therefore, permutations are sequences where order is important, whereas in case of combinations, the order is not important. Let us understand both these concepts with the help of an example.

Consider the following question:-

In how many different ways can five people sit in a row?

In this question, we have to arrange five people in a sequence. However, the order is important in solving this problem. Now, for the first spot in the row, we have five people to choose from. As we go down the line, the number of people we have to choose from will decrease by one. Therefore, we have 5 x 4 x 3 x 2 x 1 = 120 ways in which five people can sit in a row.

Now, consider this question:-

Steven drew 13 portraits for his art project. He wants to choose 8 of them for his studio art final. How many different groups of photos are possible?

If you observe carefully, this question asks about groups and not the order in which the portraits will be chosen. Therefore, this is a combination problem. Therefore, using the formula-

nCk = n! /��k! * (n–k)!

= 13! /��8! * (13–8)!

= 1287 different groups

Download our eBOOK on “How to Master GRE Probability“

Tips to master GRE Probability questions

Tip 1- Confirm that the question is on probability:

In GRE questions don’t come labeled or tagged as arithmetic or probability. It is therefore important to determine whether the problem involves probability or not. So, look for the word ‘����Dz������������ٲ�’ in the questions. If you cannot find the word ‘����Dz������������ٲ�’ embedded in the question description then look for other important and related key terms such as- event, outcome, random selection.

Consider the following example-

1) A fair dice is tossed. Event A is defined as getting a prime number and event B is defined as getting an even number. Which one of the following is/ are true?

(i) P(B)= 1/2

(ii) P(A n B)= 0

(iii) P(A) + P(B)= 1

(iv) P(A n B)= 1/6

(v) P(A)= 4/6

Notice in this example question that the word probability has not been used. But as you can observe the word event appears in the question. This means that the question involves probability.

Tip 2- Memorize the formulas:

If you want to have a good Quants score then memorizing the key formulas in probability is important. You should not be wasting time trying to figure out what the formula was as you don’t get a lot of time to solve the questions. Our experience with GRE aspirants shows that top scorers actually spend more time learning formulas and concepts rather than practicing sums.

These are some of the important probability formulas you can memorize and keep in handy-

The probability of an event = Favourable number of event number of total outcomes

Odds in favor of an event = Number of favorable outcomesNumber of total outcomes

Odds against an event = Number of unfavorable outcomesNumber of total outcomes

Download our eBOOK on “GRE Math Formula Book for a Perfect 170“

Tip 3- Practice a lot:

There is no shortcut to success. And that is true even for GRE Quants. If you are dreaming of higher Quants score, your dream will come true with thorough practice. Data shows that difficulty level of questions on probability is usually medium to high. So, make sure that you practice GRE questions which are a little tough. You must practice advanced GRE questions of level 4 and level 5.

In order to secure a higher percentile in GRE Quants, it is important that you gain enough understanding of probability concepts. In fact, achieving a perfect 170 in Quants is not a very difficult task. What you need is some advice and guidance regarding how to prepare and avoid some common mistakes that GRE test takers often make. You can check out our video on Quant Shortcuts! (Probability)

We hope this article provided you with smart ways to crack GRE Probability questions. All the best!

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• Algebra
• Arithmetic
• Geometry
• Applied Mathematics
• Data Interpretation

For most engineers, after doing more complex mathematics for four years, questions based on the concepts of algebra or arithmetic are easier to tackle compared to Data Interpretation or Geometry. This is simply because they are out of practice.

But why should you focus on Data Interpretation particularly?

If you aim to hit a 160+ in Quant, then is important to spend enough time learning and practicing Data Interpretation because a minimum of 4 or 5 questions can be expected in each Quant section. This means that DI carries a 20% weightage in the Quant section.

Makes it pretty important to get right to a high score, don’t you think?

Structure of a GRE Data Interpretation Question:

In a DI question, you will face one form of pictorial data such as a bar chart (either horizontal or vertical), pie charts, line graphs or tables, followed by 3 or 4 questions on the same. The question types could be multiple-choice or numeric entry types.

Throughout this post, you’ll see all the different types of examples!

For example,

The line graph shows the ratio of imports by a company to the number of exports from that company over the period 2000–2006.

Q. In which year, was the imports to the exports of the company greater than 0.3 and less than 1.1?

Indicate all such years.

Choices:

1. 2000
2. 2001
3. 2002
4. 2003
5. 2004
6. 2005
7. 2006

Ans: The correct answers are options A, B, C, and F

From the graph,

Ratio of value of imports to exports in 2000 = 0.65

In 2001, ratio of value of imports to exports = 0.85

In 2002, ratio of value of imports to exports ��= 0.35

In 2003, ratio of value of imports to exports = 1.25

In 2004, ratio of value of imports to exports = 1.4

In 2005, ratio of value of imports to exports = 0.95

In 2006, ratio of value of imports to exports = 1.55

The correct answers are ��A, B, C, and F

How can you master Data Interpretation? Learn how in just a few key steps, through an ebook that’ll make you an expert! Download your FREE eBook now!

Some common GRE Data Interpretation mistakes to look out for:

1. Misinterpreting data due to impatience/over-confidence
2. Time management
3. Lack of practice
4. Overlooking details

Since there are 20 questions in each Quant section and you have only 35 minutes per section, you cannot spend more than 2 minutes on any of the questions.

So, within 2 minutes you need to assimilate data, interpret it correctly, and answer questions that follow. This can be quite a task if you are not familiar with reading data.

So, how do you pick out the relevant information and do it for 2 minutes?

��How to Approach a GRE Data Interpretation Question:

First, don’t be alarmed when you see data that apparently seems complicated. There is a method to the madness.

There are important techniques that you can follow that will help you figure out the right answers.

#Tip 1: Look for tell-tale signs:

Your graph may look complicated but there are some key points that you should note down as soon as you see a graph, like the axes of the graph, the units of measurement, and the title of the graph. It is important to pay attention to the title because you will be able to pick out crucial facts like what the data represents.

For example,

Annual Percent Change in the Dollar Amount of Sales at Five Retail Stores from 2006 to 2008

Q. At Store P, if the dollar amount of sales was $800,000 for 2006, what was the dollar amount of sales for 2008?

1. $727,200
2. $792,000
3. $800,000
4. $880,000
5. $968,000

Ans: The correct answer is Option B, $792,000.

Explanation from our SFAs:

Since it is mentioned that the sales of Store P was $800,000 in 2006, we can use that to calculate the��respective rise in percentages for 2007 and 2008.

According to the figure, the percentage increase for 2007 is 10%. So, we will calculate it as 110% of the base amount, which is, 110% of $800,000 = $880,000.

Similarly for 2008, the percentage decrease is 10%. The calculation is 90% of $880,000 = $792,000.

��ɫ����Ƶ Pro Tip: Always ensure you check the base amounts before doing calculations like these. A percentage increase of 10% in 2007 followed by a percentage decrease in 2008 does not mean the answer is the same as the initial amount because the base amounts for 2007 and 2008 are different.

Thus, “Annual percent change in the dollar amount of sales at five retail stores from 2006 to 2008” tells us that the data represents “percent change” rather than absolute percentages.

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#Tip 2:��The Art of Being Approximately Accurate:

The GRE is meant to test your presence of mind apart from your reasoning skills. This means that you will encounter some challenging numbers and calculations. Before you jump to the online calculator and start crunching numbers, wait! Round up decimals and approximate to the nearest whole numbers so that you know the ballpark figure. For example, 27% of 153,000 could be rounded off to 25% of 150,000 so that you can eliminate far-fetched options.

��ɫ����Ƶ Pro Tip: Beware and use this sparingly. If the answer choices are pretty close to one another, it is best to skip this step and directly calculate your answer. If your range includes two options then it makes it necessary to actually calculate the answer causing you to waste more time.

#Tip 3:��Say What?

I cannot emphasize this enough because this is one of the common reasons behind students scoring a 165+ but not a 170. Read the question carefully and understand clearly what you’ve been asked to answer. Many times, the answer choices have two right answers but to two different questions. If you haven’t read the question correctly because you were in a time crunch, you will choose the other answer and hence lose out.

For example,

Q. If 3000 copies of a book are published and the paper cost on them amounts to $ 9000, what should be the selling price of a book so that the publisher can earn a profit of 25%?

Choices:

1. $ 3.4
2. $ 12.5
3. $ 37, 500
4. $ 10
5. $ 2.5

Ans: The correct answer is Option B, $12.5.

Explanation from our SFAs:

Let the SP of 3000 books be $ X.

To earn a profit of 25%, SP = CP + 25% of CP = 125% of CP

From the given pie-chart,

paper cost = 30% of CP

$ 9000 = 30/100 * CP

CP = 9000 × 100 / 30 = $ 30,000

SP = 125% of $ 30,000

= 125 / 100 × 30,000

= $ 37,500

Therefore, SP of one book = 37500 / 3000 = $ 12.5

For a pie-chart question, bear in mind that the whole pie is split into parts of a whole either using percentages or degrees.

The following conversion can be used to calculate the value of the sector based on what is given in the pie-chart.

Value of the sector = (Percentage / 100) * Total value

Value of the sector = (Central angle of the sector / 360)* Total value

For this question, after a quick scan, if you assumed the “paper cost” to be 108%, then you would choose the first option i.e., $ 3.4, which is incorrect.

Confused about the different questions in GRE Quant? We’ve made it super simple and easy! Get your guide to the different types of GRE Quant Questions you can expect! Download your FREE guide now!

#Tip 4:��Develop an Eagle Eye for Charts:

Interpreting data quickly is a skill that comes with practice. Make sure you practice different kinds of charts and questions. This will help you overcome most of the problems mentioned above. And since DI is a sure-shot question, it pays to perfect your techniques.

��ɫ����Ƶ Pro Tip: Practice questions where there is more than one type of chart and you have to correlate data. This will help you understand the relationships between different types of data and the variables connecting them. This helps you understand data analysis as a whole.

For example,��Profit and Turnover of Company PQR Over the Period 2004-2008

Q. If the turnover in the year 2008 is decreased by��15% and the ratio of the profit and operating cost remains the same, what will be the profit in the year 2008?

Note: Turnover – Operating Cost = Profit

1. 0.48
2. 0.408
3. 1.28
4. 1.088

Ans: ��The correct answer is Option B, 0.408.

Explanation from our SFAs:

Given data: Total turnover, Total profit, Ratio of profit & Operating cost are the same & ��Turnover – Operating cost = Profit

Total profit = Profit in (2004 + 2005 + 2006 + 2007 + 2008)

$1.44 million = $0.36 million + $0.288 million + $ 0.192 million + $0.12 million + Profit in 2008

Therefore, profit in 2008 [P(2008)] = $ 0.48 million ———-> (1)

But, Turnover in 2008 = 1.76
Operation cost of year 2008 [OC(2008)] = Turnover – Profit = 1.76 – 0.48 = 1.28 ——> (2)

Therefore, P(2008) / OC(2008) = 0.48 / 1.28 ———–> (3)

Given that, turnover in the year 2008 is decreased by 15%.

So, the New turnover = 1.76 – (15% * 1.76) = 1.76 * 0.85 = 1.496 —–> (4)

Now, if the new profit and operating cost are P1 & OC1 respectively, then according to the question,

P(2008) / OC(2008) = P1 / OC1

Therefore, P(2008) / OC(2008) = P1 / OC1 = 0.48 / 1.28

Since, we have to calculate profit, OC1 = P1 * (1.28 / 0.48) = P1 * 2.67 —–> (5)

Operating cost = Turnover – Profit

Therefore, New turnover = P1 + OC1

1.496 = P1 + P1*2.67 = P1 * 3.67

Therefore, P1 = 1.496/3.67 = 0.408

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#Tip 5: Befriend the Pen & Paper:

When you are practicing DI questions, always use a pen and paper to jot down the important points mentioned in the question as well as minimum and maximum values, reference values based on which the graph is drawn, etc. With time, you will become accustomed to picking out these values as soon as you see a graph.!

To summarize, these are some of the points to remember:

1. Spend only 15 to 30 seconds analyzing the graph and jotting down key information. Don’t spend more than 2 minutes per question.
2. The answer is literally staring at you in the face in the graph. So, don’t feel disheartened. Mark it for verification and get back to it, if you are unsure of the answer.
3. There is no negative marking in GRE. Approximate and make an intelligent guess, but make sure you answer. You may be lucky!
4. The devil is in the details. Make sure you pay attention to the axes, the units asked the question, concepts used like percent changes or sector to percentage conversion etc.
5. Try to practice as many questions as you can so that you feel, “Oh, I’ve seen this kind of question before,” during the exam!

Comment and let us know how we can help you with your Quant difficulties!

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