Math Things to Memorize: the Combinations Formula

For some questions on the GMAT, you will need to know the Combinations formula.

Combinations formula = n! / k! (n-k)!

n = the bigger number (what we’re choosing from) 
k = the smaller number (how many we’re choosing)

Check out this question from Math Revolution:

How many committees can be formed comprising 2 male members selected from 4 men, 3 female members selected from 5 women, and 3 junior members selected from 6 juniors?

A. 900
B. 1200
C. 1500
D. 1800
E. 2400

We could almost rephrase this as three different questions:

How many ways to choose 2 from 4?
How many ways to choose 3 from 5?
How many ways to choose 3 from 6?

Let’s start with the first question: how many ways to choose 2 from 4? 

4! / 2! (4-2)!
4! / 2!2!
(4 x 3 x 2 x 1) / (2 x 1)(2 x 1)
24/4 = 6

But if we counted this and said the four males were ABCD, we could just list AB, AC, AD, BC, BD, and CD, and you can see it also equals 6.

Anyway, finding the other two:

How many ways to choose 3 from 5?

5! / 3! (5-3)!
5! / 3! 2!
We can cancel out 3! from both numerator and denominator.
5 x 4 / 2
20/2 = 10

How many ways to choose 3 from 6?

6! / 3! (6-3)!
6! / 3! 3!
We can cancel out one 3! from both numerator and denominator.
6 x 5 x 4 / 3 x 2
120 / 6 = 20

Back to our original questions:

How many ways to choose 2 from 4? 6
How many ways to choose 3 from 5? 10
How many ways to choose 3 from 6? 20

Now, multiply all those numbers together! :)

6 x 10 x 20 = 1200

The correct answer is (B).

How to Strengthen an Argument in GMAT CR

There’s several ways to strengthen an argument. We can add new evidence, support the existing evidence, or perhaps even give a tidbit that might show an assumption is likely. Strengthen questions can be trickier than Assumption or Evaluate, because you can’t just hone in on the Conclusion and ignore or devalue everything else. With Strengthen (and Weaken) you have to take into account the ENTIRE picture. There might be 2-3 ways to strengthen, and the correct answer is not the first one that comes to your mind!

Let’s look at an argument:

The government is being urged to prevent organizations devoted to certain views on human nutrition from advocating a diet that includes large portions of uncooked meat, because eating uncooked meat can be very dangerous. However, this purported fact does not justify the government’s silencing the groups, for surely the government would not be justified in silencing a purely political group merely on the grounds that the policies the group advocates could be harmful to some members of society. The same should be true for silencing groups with certain views on human nutrition.

Which one of the following principles most helps to justify the reasoning in the argument?

(A) The government should not silence any group for advocating a position that a significant proportion of society believes to be beneficial.
(B) The government ought to do whatever is in the best interest of society.
(C) One ought to advocate a position only if one believes that it is true or would be beneficial.
(D) The government ought not to silence an opinion merely on the grounds that it could be harmful to disseminate the opinion.
(E) One ought to urge the government to do only those things the government is justified in doing.

Let’s break this down:

Evidence: Gov’t urged to stop anti-meat groups

Conclusion: Gov’t not JUSTIFIED in SILENCING them

Additional Evidence: Gov’t shouldn’t silence just because harmful to “some” — similarly, shouldn’t silence nutrition groups

The author is assuming that these “certain views” on nutrition might also be harmful to “some” members of society, so the gov’t might want to silence them. To strengthen, perhaps we could get additional evidence on the potential dangers of these pro-nutrition groups? Or perhaps brand-new info that shows why the gov’t is not justified in silencing the anti-meat groups?

PREDICTION: Certain nutrition groups also harmful and gov’t doesn’t silence them. Anything that shows the gov’t lacks “justification” to silence anti-meat. Perhaps a good thing about anti-meat? (I admit, I’m reaching a bit.)

We can have our Prediction but ALSO be a little more open to what the answer choices will bring with this one. :)

(A) The government should not silence any group for advocating a position that a significant proportion of society believes to be beneficial. (it’s not really about the proportion of society)
(B) The government ought to do whatever is in the best interest of society. (“best interest” doesn’t really relate to meat or nutrition groups or the idea of justification)
(C) One ought to advocate a position only if one believes that it is true or would be beneficial. (this kind of general “one” talk pops up a lot on LSAT questions, but not really on the GMAT. It’s wrong.)
(D) The government ought not to silence an opinion merely on the grounds that it could be harmful to disseminate the opinion.(more on topic, let’s keep for now)
(E) One ought to urge the government to do only those things the government is justified in doing. (don’t love the “one” talk, but it’s more on topic, so let’s keep for now)

Now that we have identified the Final Two, let’s take a closer look at which one is more focused on the specifics of the argument:

(D) The government ought not to silence an opinion merely on the grounds that it could be harmful to disseminate the opinion.
(E) One ought to urge the government to do only those things the government is justified in doing.

Let’s look at the conclusion again:

This purported fact (uncooked meat = dangerous) does not justify the government’s silencing the groups.

It’s really about the government’s actions, not really what the PEOPLE should be. Like (C), choice (E) has that weird “one ought” language. With CR, we aren’t really here to make moral judgments. We want to strengthen the idea that the GOV’T is not JUSTIFIED.

If we rephrase (D) it says: “gov’t shouldn’t silence opinions that could be alarming.” This re-states and re-energizes the idea that “gov’t shouldn’t silence just because harmful to ‘some'”.

The correct answer is (D).

GMAT DS: The Statements Cannot Contradict Each Other!

 

 

 

 

 

 

A few good takeaways from this question:

-we can always rewrite integers as their factors, so we can say 9 = 3^2
-we can multiply across inequalities freely (without flipping the sign) when we know the variables are positive
-if a smaller integer is LARGER than a bigger integer and they both have unknown exponents, then the exponent of the smaller integer is obviously “making up” for the difference in value and must be larger than the bigger integer’s exponent

The answer is (D).

The flaw with the question: The statements are incompatible. It cannot be that x = 0 and y = 2, and Statement (2) is true. Because if we plug in x = 0 and y = 2, then Statement (2) reads:

1/16 > 16/9
1/16 is not larger than 16/9

So, while there’s good takeaways here, the fact that the statements contradict one another do not make this a great GMAT question. It is not an official GMAT problem for this exact reason.

Here’s a little more scratchwork, if you’re curious!

Backsolving Problem Solving “Work” Questions

One way to do these type of question is to Backsolve, or essentially “try out” the answer choices. Let’s look at a problem:

Working together, each at his or her own constant rate, Jeff and Ashley painted their apartment in 6 hours. Working at his constant rate, Jeff could have painted the whole apartment in 10 hours. How many hours would it have taken Ashley, working at her constant rate, to paint the apartment?

A. 4
B. 12
C. 15
D. 16
E. 20

Let’s say it takes Ashley 15 hours (choosing answer choice (C)). Ashley would have a 1/15 rate and Jeff’s rate is 1/10 each hour. Working together they would do 1/15 + 1/10 each hour, or 2/30 + 3/30 = 5/30 = 1/6 in one hour. Then, yes, it does make sense that together the would do the job in 6 hours.

We got lucky here that (C) ended up being the correct answer, but if our answer had not matched the given information, then we probably could have discerned the correct answer based on how “too big” or “too small” we were.

While not required, sometimes we forget how useful leveraging the answer choices can be! :)

Possessives on GMAT Sentence Correction

In the latter years of the Great Depression, colleagues of Franklin Roosevelt’s argued that his proposed Social Security Act has a good chance of succeeding due to its strong bipartisan support, sound fiscal plan, and widespread electoral approval.

A. Roosevelt’s argued that his proposed Social Security Act has
B. Roosevelt’s argued that his proposed Social Security Act had
C. Roosevelt’s have argued that his proposed Social Security Act had
D. Roosevelt argued that his proposed Social Security Act had
E. Roosevelt had argued that his proposed Social Security Act has

In the first phrase, we get a great clue about what time period these colleagues lived: the “latter years of the Great Depression.” This is obviously in the PAST, so we need the action that the colleagues did to be a past tense verb. Answer choice (C) is present perfect tense (used to describe something that started in the past and CONTINUE to the present. But these colleagues are dead now, so how can they still be arguing?)

In (E) we have Past Perfect, which is a kind of past tense, BUT we only use this tense to describe an event that occurred before a Simple Past Tense event. In (E), “has” is not past tense, and even if it were, the meaning wouldn’t make sense. The colleagues didn’t argue BEFORE the Social Security Act had a chance of success.

So, now we know it has to be (A), (B), or (D). Since (A) also uses the word “has” and we discussed this in (E), we can eliminate this, too.

Let’s focus on the differences of the Final Two:

(B) In the latter years of the Great Depression, colleagues of Roosevelt’s argued that his proposed Social Security Act had a good chance of succeeding due to its strong bipartisan support, sound fiscal plan, and widespread electoral approval.

(D) In the latter years of the Great Depression, colleagues of Roosevelt argued that his proposed Social Security Act had a good chance of succeeding due to its strong bipartisan support, sound fiscal plan, and widespread electoral approval.

Wow! Only one difference. :) So what do we know about the use of possessives? When a noun turns into a possessive, it becomes a modifier.

EX: Jill won the soccer game.
Meaning: Jill is the one who won.

EX: Jill’s team won the soccer game.
Meaning: The TEAM won, and “Jill’s” just describes the team.

So in (B), if “Roosevelt’s” is now a modifier, the question becomes, what is it modifying? The only logical option is “colleagues.” So the meaning of (B) is “colleagues of Roosevelt’s colleagues.” Um…what? That’s redundant.

The correct answer is (D).

The Best Strategy for GMAT Evaluate the Argument Questions

With an Evaluate the Argument question, we have to keep in mind that it’s aaaaalll about that Conclusion! This isn’t like a Weaken or a Strengthen in which some tiny piece of evidence will twist around and be part of the correct answer, unexpectedly.

Evaluate questions are easy if you keep in mind that you are here to evaluate the Conclusion sentence and that’s it! So let’s break one down:

The US government has recently taken an initiative to collect and publish information on the salaries of graduating students from colleges. The salaries of the students in their first year after graduation will be published for all colleges and subject fields the colleges offer. The idea is to help the students make more informed choices about the college and the field that they choose. While the intentions are good, the results might just be the opposite. Students who pick their field based primarily on post-graduation salaries, as opposed to passion for a field, will, in all likelihood, struggle in both school and career.

Which of the following options would help most to evaluate the given argument?

A) What is the number of colleges that will be covered by the government initiative?
B) Currently, what proportion of students who struggle in college also struggle in their careers?
C) Do some students currently pick their subject fields based on their passion?
D) Are there currently any good websites providing average salaries data for the students?
E) How will the government ensure that the data published on the salaries of the students is not biased against certain colleges?

Evidence: US govt gets $$$ info (to help students choose).

Conclusion: Students who choose for $$$ / not :inlove: will :cry: :cry: :cry:

Ridiculous emojis aside, we can see the scope of this conclusion is about what the RESULT will be when students base their CHOICE on $$$$$. Definitely the correct answer needs to match that scope!

Let’s look at the scope of each answer choice:

A) What is the number of colleges that will be covered by the government initiative? (the number affected is not part of the conclusion’s scope)
B) Currently, what proportion of students who struggle in college also struggle in their careers? (better than A, but still not great, but let’s keep it for now)
C) Do some students currently pick their subject fields based on their passion? (decent, related to students, let’s keep it)
D) Are there currently any good websites providing average salaries data for the students? (what the heck?! this has absolutely nothing to do with the students and their choices)
E) How will the government ensure that the data published on the salaries of the students is not biased against certain colleges? (who cares about the data; we’re interested in the students’ choices)

The Final Two here are (B) and (C). Let’s answer these hypotheticals:

B) Currently, what proportion of students who struggle in college also struggle in their careers?

Let’s say 100% who struggle in college struggle in careers. Or let’s say 0% who struggle in college also struggle in their careers. This has no bearing on whether students who pick $$$ over passion will succeed.

C) Do some students currently pick their subject fields based on their passion?

Let’s say yes, some students DO pick their major based on :inlove: ; it doesn’t have a huge impact. BUT, what if NO students choose for passion? Well if none choose for passion, and they ALL choose for $$$$, then the author’s argument is greatly weakened!!! In that case, there wouldn’t even be a dichotomy — no choice at all! They ONLY choose for $$$. So how could choosing for $$$ over :inlove: even be possible?

Because one way to answer (C) has a major impact on the Conclusion, this is the correct answer. Playing “Devil’s advocate” for each “side” of the Evaluate answer choice can help you see which one is most relevant to the Conclusion.

 

Distance and Rate Problems in Data Sufficiency

This is a “Value” DS question. The Value we are looking for is the time it took Bob to finish. Let’s look at the specific wording:

How long did it take Bob to complete the race?

(1) If Bob were 2/3 faster, his time would have been 3 hours.
(2) Bob’s average speed was 30 miles per hour.

Given information: We know that Distance = Rate x Time, so if we knew Distance/Rate, then we could find the Time. We could think of this question as asking, what is the ratio between Bob’s Distance and Bob’s Rate?

Need: Distance/Rate

Statement (1):

We know D = R x T.
This says that (5/3)D = R x 3.

Let’s simplify:
(5/3)D = 3R
5D = 9R
D/R = 9/5

Sufficient! We found the ratio we were looking for!

Statement (2):

Average Speed = Total Distance / Total Time
30 = D/T

Unfortunately, we cannot find T. This is insufficient.

The answer is (A).