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).

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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! :)

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).

Practice with Functions Questions on GMAT Quant

For which of the following functions is f(-1/2) > f(2)?

A. f(x) = 3*x^2

B. f(x) = 3*x

C. f(x) = 3 + x^2

D. f(x) = 3 + 1/x

E. f(x) = 3/x^2

Students get confused on these type of questions sometimes because we aren’t generally used to plugging into answer choices TWICE. But that’s what you have to do for each answer choice. You have to plug in -1/2 and then also plug in 2, and see is the result when you plug in -1/2 is greater than the result when you plug in 2. This will only be true for ONE answer choice in this Magoosh question.

A. f(x) = 3*x^2

If x = -1/2, then f(x) = 3/4.
If x = 2, then f(x) = 12.

Is 3/4 > 12? Nope! Cross this one off.

B. f(x) = 3*x

If x = -1/2, then f(x) = -3/4.
If x = 2, then f(x) = 6.

Is -3/4 > 6? Nope!

C. f(x) = 3 + x^2

If x = -1/2, then f(x) = 3.25
If x = 2, then f(x) = 7

Is 3.25 > 7? No!

D. f(x) = 3 + 1/x

If x = -1/2, then f(x) = -2
If x = 2, then f(x) = 3.5

Is -1/2 > 3.5? Nope.

E. f(x) = 3/x^2

If x = -1/2, then f(x) = 12
If x = 2, then f(x) = 3/4

Is 12 > 3/4? Yes! We finally have an answer!

Even though it’s a PS question, it kind of has that Data Sufficiency vibe in which you’re testing cases. Here we had to “test out” each of the answer choices and see which one gave us the relationship we were looking for.

Refreshing GMAT Quant: Parallel Lines and Angles

Let’s look at this GMath problem to take a little time to review all the relationships that angles and transversal lines can create!

 

 

 

 

 

 

 

 

If line AC bisects angle BCD, what is the measure of angle ADC?

(A) 20 degrees
(B) 30 degrees
(C) 40 degrees
(D) 45 degrees
(E) 50 degrees

The lines/angles and triangles rules to refresh for this one:

-When two parallel lines are cut by a transversal, corresponding angles are equal
-Vertical angles are equal
-Supplemental angles sum to 180 degrees
-All the angles around one point sum to 360 degrees
-All three angles in a Triangle sum to 180 degrees

If you know all those rules and are comfortable with “extending lines,” then this question isn’t too challenging. :blushing

Let’s “fill this in”!

 

 

 

 

 

 

 

 

 

 

 

 

 

The correct answer is (B).

Subjunctive Mood: Spotting “Bossy Verbs”

Despite the thousands of protests from devoted fans who demanded that he should shave off his new moustache, Freddie Mercury insisted on changing his visual image.

A. demanded that he should
B. were demanding him that he
C. demanded that he
D. had demanded him to
E. demanded for him to

“Demanded that” is one of the GMAT’s rare Subjunctive “bossy verb” structures.

The structure always goes: BOSSY VERB + that + infinitive verb (without the “to”)

Only (A) and (D) include the word “that,” and of those two, only (C) includes the infinitive verb structure without the word “to” — we don’t need this extra word “should” when we have the subjunctive form.

DEMANDED THAT he SHAVE is what we want since “shave” would be the infinitive verb “to shave” with the preposition “to” dropped.

The correct answer is (C).

More “bossy verbs” to memorize:
-require THAT
-mandate THAT
-recommend THAT
-propose THAT
-request THAT
-asks THAT
-request THAT

Similar questions to practice:
-https://gmatclub.com/forum/mandate-vs-recommend-bossy-verbs-sc-241700.html
-https://gmatclub.com/forum/the-gorton-dodd-bill-requires-that-a-bank-disclose-to-their-86342.html
-https://gmatclub.com/forum/legislation-in-the-canadian-province-of-ontario-requires-of-both-publi-79769.html
-https://gmatclub.com/forum/judicial-rules-in-many-states-require-that-the-identities-of-all-prose-115077.html