Permutations, Combinations, and Probability

What are permutations and combinations? In mathematics, they are two different things, and have specific definitions. However, you do not need to know the definitions for the SAT (if you really want to know, look at Tip #2). In everyday English, the two terms are often used interchangeably. They refer to the number of ways of doing something.

For example, there are 2 possible outcomes when you flip a coin (assuming it can’t land on its edge)

If you roll a die, there are 6 possible outcomes.

So don’t worry about the terms “permutations” and “combinations.” You can just say “outcomes,” “orderings,” or “arrangements” (which is the term you’ll usually see on the SAT).

What is probability? Well, it’s the likelihood that something will occur. In everyday life, probability is usually expressed as a percentage, and it’s often very difficult to determine.

What is the probability that our lacrosse team will make the playoffs?

What is the probability that President Obama will be re-elected?

In math (including the SAT), it’s a different matter. Probabilities are expressed on a scale of 0 to 1 (which is equivalent to 100%), and they can often be determined exactly. Each probability is a fraction equal to (number of arrangements that qualify)/(total number of possible arrangements).

Q. If a die is rolled, what is the probability that an even number will appear on top?

There are 6 possible outcomes {1,2,3,4,5,6}, and 3 of them are even, so the answer is 3/6 or 0.5.

Of course, you won’t usually encounter such an easy question on the SAT. So here are some tips to help:

Tip #1 (basic) Multiplication Rulez!

Sorry for the spelling – even nerds want to sound cool once in a while.

This tip is mostly self explanatory – multiply everything (don’t add).

Q. If a fair coin is flipped twice, what is the probability of seeing heads both times?

0.5 X 0.5 = 0.25

In other words, the probability of two events occurring in succession equals the product of the individual probabilities (P[event 1] X P[event 2]).

Q. If two dice are rolled separately, what is the probability of rolling a 3 followed by an even number?

1/6 X 3/6 = 1/12 or 0.83

Q. If a random number from 1 to 10 is chosen, and a random letter from A to Z is chosen next, how many different arrangements are possible?

10 X 26 = 260

Q. A company that has 5 salespersons wants to send one of them to New York and the other to Minnesota. How many different ways are there to choose them?

Say you pick the New York salesperson first – you have 5 to choose from. Then you’ll have 4 left over to choose for the Minnesota job. That’s 5 X 4 = 20 possibilities. Note that it doesn’t matter which state you pick first.

Tip #2 (intermediate) 2 Important Questions

On harder questions, you need to consider a couple of circumstances that affect the number of orderings.

Question A: Does order matter?

Consider the following 2 questions:

Q. A cheerleading squad has 8 members. If 2 co-captains are to be chosen, how many different arrangements are possible?

Q. A booster club has 8 members. If a President and a Vice President are to be chosen, how many different arrangements are possible?

At first glance, you might think that both questions have the same answer.  But hold on – if you choose Arlene as the first cheerleader and Pablo as the second, isn’t that exactly the same as choosing Pablo first and Arlene second? But that doesn’t hold true for the booster club. Ricky for president and Jennifer for VP certainly isn’t the same as Jennifer for President and Ricky for VP.

In the first case (cheerleaders), order doesn’t matter. For every ordering X, then Y, there will be a reverse ordering that you shouldn’t count. So you need to divide your answer by 2 (there are (8 X 7)/2 = 28 arrangements).

In the second case, order does matter, so you should count every arrangement (8 X 7 = 56 possible).

Most of the time, order will matter, and you don’t have to worry. But if you’re picking several interchangeable things from a group, you need that extra step.

Every time I’ve seen this come up on the SAT (order doesn’t matter), you are asked to choose 2 or 3 things from a group. If you have to choose 2 things, you know you need to divide the number of arrangements by 2 to get the number of unique orderings. What if you’re asked to choose 3 things from a group? Then you need to divide by 6! Why 6? Well, suppose you choose A, B, and C. There are 3 X 2 X 1 = 6 possible ways to order them (ABC, ACB, BAC, BCA, CAB, CBA). And there will be six ways to order any subgroup (such as ADE, etc.), and you only want to count each subgroup once.

Just in case you ever have to choose 4 things from a group, remember to divide by 24 (4! = 4 x 3 X 2 X 1).

By the way, this is the difference between the mathematical terms “permutation” and “combination.” If order matters, it’s a permutation.

Question B: Is there replacement?

Q. If a bag contains 4 blue marbles and 4 red marbles, and 2 marbles are taken, what is the probability that they will both be blue?

The wording must be changed to specify whether or not you can take one marble, put it back and the bag, and then choose another.

No replacement: 4/8 X 3/7 = 12/56 = 3/14

Replacement (first marble replaced before second is chosen): 4/8 X 4/8 = 16/64 = 1/4

Replacement is sometimes called repetition.

Q. A butterfly collecting club has 8 members. If a President and a Treasurer are to be chosen, how many different arrangements are possible?

Can the same person serve as both president and treasurer? If you don’t know, you can’t answer the question.

If two different people must serve, there are 8 X 7 = 56 arrangements. If someone can hold both offices (i.e. repetition is possible), it’s 8 X 8 = 64.

Usually, there won’t be replacement on the SAT. Just watch out for it.

Tip #3 (advanced) Do Restricted Things First

On all of the questions we’ve looked at so far, the situation involved unrestricted events. You could pick any member as president, or any marble first, etc. But on harder SAT questions, you will be presented with restricted choices.

Q. Ahmad, Barbara, Chen, DeShawn, and Enrique will each read a poem in English class during the coming week. One student will read a poem on each day of the week, and only Ahmad or Chen will be permitted to read on Wednesday. How many arrangements are possible?

The trick here is to choose Wednesday’s speaker first, since Wednesday is restricted. You can choose either of 2 students (A or C) on that day. Now you can choose in any order you like, since the other days aren’t restricted. If you go from early to late, you’ll have a choice of 4 students for Monday (since one was already chosen), 3 for Tuesday, 2 for Thursday, and 1 for Friday. So the answer is 2 X 4 X 3 X 2 X 1 = 48.

Q. Ahmad, Barbara, Chen, DeShawn, and Enrique will each read a poem in English class during the coming week. One student will read a poem on each day of the week, and Ahmad must read on Tuesday, Thursday, or Friday. How many arrangements are possible?

Now Ahmad is restricted, instead of Wednesday. So we do a person first instead of a day. There are 3 possible days for Ahmad, and then there will be 4 days left to choose for Barbara, etc. 3 X 4 X 3 X 2 X 1 = 72.

Finally, geometric probability questions are quite easy. If an object is tossed into Area A, what is the probability that it will land in Area B (a smaller area within Area A)? Answer: Area B/Area A.

If you study hard, there is a high probability that you’ll do well on your test!

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Graph Tricks for the SAT

Most of the graphs you’ll see on the SAT are linear (straight line) graphs. For any other shape, you should concentrate on one point at a time; don’t get hung up trying to “understand” such graphs.

Here are three tricks to make linear graphs easier to deal with:

1) Slope Trick

Q. What is the slope of the line that contains the points (2,7) and (4,5)?

First of all, on the SAT, the word “line” means “straight line.” You might say “I see a curved line,” but the SAT won’t.

In math class, you learned that the slope = change in y/change in x, or rise/run. Then you learned the Point Slope Form to determine the slope:

y y1 = m(x x1)

You may or may not find that daunting. In any case, there’s a much easier way than memorizing and solving an equation with 5 variables. First, write the 2 points vertically:

(2,7)

(4,5)

…and then just write the changes in y and x to the right and left of those points:

(2,7)

+2 (4,5) -2

That is, the y values went from 7 to 5, which is -2, and the x values went from 2 to 4, which is +2. So the change in y/change in x is -2/+2, or -1.

Yes, the logic is the same as that of the formula, but it’s a lot harder to mess up.

2) Distance Trick

This is another trick in which common sense beats algebra. For the distance d, you were taught this formula:

d = √[(y2 – y1)2 + (x2 – X1)2]

You were probably also taught that this formula is based on the Pythagorean Theorem (a2 + b2 = c2), which is a much easier formula to use. The trick here is just to substitute the change in y for a, and the change in x for b. Then solve the Pythagorean Theorem for c, which is the distance.

Q. What is the distance between the points (7,6) and (-1,5)?

First, get the change in y and change in x as you did for the first trick:

(7,6)

-8 (-1,5) -1

Use -1 for a and -8 for b (actually, it doesn’t matter if you switch them.  You can also ignore the negative signs since you’re going to square them anyhow).

a2 + b2 = c2

12 + 82 = c2

1 + 64 = c2

√65 = c (the distance)

In the diagram, you can see how the change in y and change in x are the legs of a right triangle for which the hypotenuse is the distance.

3) “Human Graphing Calculator” Trick

A Human Graphing Calculator (HGC) can look at a graph and immediately tell you the approximate formula. And if you provide a formula, an HGC can quickly show you the graph.

Maybe HGC’s really exist. If so, they have a pretty freaky gift. But for linear graphs, you can be an HGC. You probably already have the tools you need.

Remember, any straight line can be expressed by the formula y = mx + b, where m is the slope, and b is the y-intercept. The y-intercept is easy – that’s where the line crosses the y-axis (where x = 0). And you should know what different slopes look like. If the slope is zero, the line is horizontal. The line goes up (from left to right) if the slope is positive; a line with a negative slope goes down. You can estimate the magnitude of the slope by how steep the line is. If it rises steeply, the slope is greater than 1. If it falls slowly (shallowly), the slope is between 0 and -1.

That’s it! If you’re given y = -3x + 5, you know the line will fall steeply and intercept the y-axis at 5. So, when you see a problem involving a straight line graph, you should be confident.

College Rankings

Scores of websites, books, and magazines offer college ranking lists, which they update each year.  Some rank according to academic prestige; others offer rankings based on quality of life on campus, while some lists are comprehensive. You can even find rankings based on perceived attractiveness of the female or male students, or which schools have the best radio stations.

The ranks that most colleges, parents, and students care the most about are the ones that concern  academic prestige. After all, we attend college to get a good jobs in the fields of our choice, and we want potential employers to be impressed by the colleges we attend.

Statue of John Harvard, founder of Harvard Uni...

Image via Wikipedia

The questions students and their parents ask most often are: 1) How accurate are the rankings? 2) How seriously should we take them? There are many articles on the web about the first question, so I will merely summarize: the rankings are fairly accurate. The same college may have rankings that are as far as 15 places apart on two of the major surveys. That’s really not so surprising, since different methods are used for various surveys.

For example, US News and World Report explains: “The indicators we use to capture academic quality fall into a number of categories: assessment by administrators at peer institutions, retention of students, faculty resources, student selectivity, financial resources, alumni giving, and (for National Universities and National Liberal Arts Colleges) high school counselor ratings of colleges and “graduation rate performance.” Some of these factors will be more important to some students than others.

So I wouldn’t get too hung up over whether Harvard or Princeton gets this year’s top ranking. But a prospective employer will probably be more impressed by a degree from the #15-ranked school, as opposed to #25.

Now on to the second question. Obviously, a college’s ranking shouldn’t be the only factor you consider when deciding whether to apply to and/or attend. Let’s consider some other factors, most of whose importance will vary from student to student.

1) $Money$ Given the current state of the economy, this will be the most important factor for many families. Many advisors admonish students that it would be a mistake to rule out private universities on the basis of cost, explaining that attractive financial aid packages can make these schools as affordable as public ones. Colleges are now required to include cost calculators on their websites, which allow you to determine how much money you’ll need, based on your family’s income. However, be aware that a college can modify its financial aid policy from year to year. I personally know two students who were forced to transfer after much of their financial aid was not renewed for their sophomore years. Furthermore, public schools are usually still a lot cheaper after you factor in financial aid.

2) Location This important factor is mostly self-explanatory. However, if your choice of schools in a particular area is too limited, you may need to be flexible. Also, be aware that two colleges in the same area can have very different campuses and external settings (e.g. within a city vs. a few miles away).

3) Size (student body) Many students all but ignore this factor, but I’ve placed it at #3 because it can have a huge bearing on your education and lifestyle. True, a large college offers diversity of professors and courses, but a small school offers small, intimate classes and a tightly knit student body.

4) Academic Programs If you know what field you want to study, you can apply to schools with strong programs in that area. If you decide later, or change your mind, you may end up wanting to transfer.

5) Safety What is the neighborhood like? How effective is campus security?

 

Flower Girl & Toy Gun

Image by greenmelinda via Flickr

 

6) Personal Preferences These include sports, religious affiliation, single-sex colleges, housing, social life, etc. Students may also opt for colleges which have more relaxed requirements for a degree, so they can take more electives.

These factors, along with others I haven’t thought of, may lead you to apply to less competitive colleges. Some students may even prefer a less competitive college, so the coursework won’t be as rigorous.

In any case, you should visit schools that you are seriously interested in attending. To be sure, I’ve known students who have been rejected by all of their top choices, and have opted to attend schools that they hadn’t visited. But remember that a school that “looks” good in an article or a website might not be to your liking.

Before I leave this subject, there is one final factor to consider: Are you planning to continue on to some type of graduate education? That’s not a factor that is often discussed in this context, but it can be very important. Obviously, you want an undergraduate education that will prepare you properly (e.g. a strong pre-med program before medical school). But it’s also important to realize that prospective employers will be more interested in your graduate college than your undergrad one.

Under certain circumstances, you can actually be better off going to a less competitive school.  For example, many law schools use an admission “index” based on your GPA and LSAT score (e.g. GPA + 1/10 LSAT). So, whether you earn an undergraduate GPA of 3.5 at Harvard or at Rinky-Dink College of Comic Literature, you will have an equal chance of admission to those law schools. Of course, your Harvard education will probably prepare you better for law school.

Vocabulary-In-Context Questions on the Critical Reading Section

3 quick hints to doing well on these:

1) Come up with your own definition before you look at the choices. This is a great technique on multiple choice questions, because it helps avoid attractive, yet wrong answers. And although it takes a bit of time, it actually helps you solve questions faster overall. Sometimes you might even see your exact word in the answers.

Example text: All bus service was suspended during the snowstorm.

…”suspended” most nearly means

Avoid the temptation to peek at the answers. This is a simple sentence, and “suspended” clearly means “stopped.”

(A) hung
(B) adjourned
(C) dissolved
(D) banned
(E) damaged

Answer (B). This should be a slam dunk if you know the word “adjourn.” Otherwise, use process of elimination. Don’t let fear of a word that you don’t know lead you to a poor decision.

Snowstorm

Image by valcomp via Flickr

If you can’t think of a good word, a phrase will do:

Example text: The presidential candidate failed to present a clear explanation of how he planned to improve the economy.

…”clear” most nearly means

If you can’t think of a single word, just say something like “easily understood.”

(A) calm
(B) transparent
(C) absolute
(D) bare
(E) understandable

The answer is obviously (E).

2) Beware the most common definition! On these questions, The College Board isn’t testing whether you’ve memorized a difficult word. Rather, they are testing whether you can recognize the correct meaning of a word that can have more than one definition.

In line 17, “sight” most nearly means

So, if you choose “vision” here, you’ll probably miss. The College Board doesn’t want to let just anyone get their questions right, and a second-grader would choose that.

 

green eyed girl..

Image by ~no bullshit~ via Flickr

 

…”company” most nearly means

(A) friends
(B) assembly
(C) objection
(D) business
(E) tribute

Avoid (A) and (D) above.

…”conduct” most nearly means

(A) contribute
(B) lead
(C) transmit
(D) behave
(E) overlook

Avoid (B) and (C) (as in “conduct electricity”) above.

3) Try your answer in the sentence to be safe. If you’ve made a mistake in your thinking, you can catch it here. And if your choice doesn’t fit grammatically, you’ll know it’s wrong.

Example text: Although her advisor made some convincing counter-arguments, Hyun Sook would not allow her resolve to be compromised.

…”compromise” most nearly means

The word “compromise” has an easier definition (to settle or agree by making concessions) and a harder one (to weaken or lower). If you don’t know the more obscure definition, you might really think the answer has to do with the common one, and like answer (A). But when you try it in the sentence, it just doesn’t fit (“allow her resolve to be agreed”), so it must be incorrect.

(A) agreed
(B) combined
(C) weakened
(D) consisted
(E) squeezed

Example text: The practice here at the Green Monkey Lodge is to remove one’s shoes before eating.

…”practice” most nearly means

(A) rehearsal
(B) career
(C) repetition of
(D) convention
(E) constitutional

Did you get (D)? Here, “practice” means something like “the thing a person does.”

SAT Tutoring and Morality

To begin with, let me reiterate that I have run an SAT prep company for 23 years.  Obviously, I believe that taking an SAT course is beneficial, but I’m sure I have some bias.  However, since SAT prep is such a big business, I think it’s reasonable to assume that most people agree that it’s helpful.

I discovered many years ago that, not only is private SAT tutoring more effective than group instruction; it is vastly so.  I found that score increases for my private students dwarfed those of group students, and I have offered only private tutoring since.  Of course, making that decision also meant that I’ve made less money than I would have.

So I chose quality over profit. How altruistic of me! But hold on.  Private tutoring is more expensive for the student, even if the teacher earns less per hour.  And that means that only families that can afford private tutoring have access to the largest score increases. Is that unfair?

 

Money!

Image by yomanimus via Flickr

 

My best answer is: sort of. I’ve thought about this issue for a long time, and there are all sorts of easy rationalizations that can cloud one’s thinking. “I’m helping students. I can’t spend my time thinking of those I can’t help.”  “That’s how the world works.” And there’s always “if I don’t do it, someone else will.”

But it isn’t hard to find the fallacies in any of those statements. So let’s look a little deeper.

First of all, I want to make it clear that I don’t think there’s anything wrong with SAT prep per se. When SAT teaching as a business took off in the 1980’s, there was plenty of backlash.  Part of that negativity arose from The College Board/ETS’s long-standing claim that the SAT couldn’t be taught. When change is afoot, complaints are never far behind. But clearly the SAT can be taught, just as the MCAT, LSAT, or GMAT can, and you don’t hear a lot of complaining about prep courses for those tests. But in those days, a lot of students simply lacked access to effective SAT courses, and they found themselves at a disadvantage.

Today, a student can easily find an SAT course, regardless of where he lives.  Online courses are an option. Some students rely only on prep books.  Very few go in “cold.” So I think we can put “prep is evil” thinking to bed. If everyone has access, and it’s effective, what could be wrong?

No – the question isn’t whether SAT teaching is unethical; it’s whether more effective, expensive teaching is. And that’s exactly what I’ve been offering for over two decades. To be sure, not every student we’ve taught has made a mammoth score increase. But our average increase is much larger than that of other SAT teachers in the area (brag, brag!), and it is easy to argue that we’re merely giving an advantage to those who have the bucks to spend.

As I’ve already mentioned, I don’t have a definitive, self-serving answer to this question. I am motivated by offering the highest quality education that I can, and I have employed a half-baked solution to this moral dilemma.  Yes – our tutoring costs more than most SAT classrooms. But we also charge a lot less than many other private tutors do. That makes our course available to the middle class, even if it’s more of a burden that a group program would be.

It’s not a perfect solution, but it’s the best I’ve come up with. Thoughts?