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# August 2016

Question: How many unique teams of 5 players can be selected from 9 who try out?

Coaches face this kind of problem all the time. Only a certain number of players can make it on to the starting roster, and there are any number of scenarios that need to be prepared for.  What if Joey gets injured? Who would replace him?  What if Nicole were moved from defense to offense?  Should Tanya take her place?  There are also all the possible tactical set-ups to think about.

In the above math problem, there are a lot of different player combinations to consider. Fortunately, counting them can be done very quickly and easily.

Permutations or Combinations?

The first step in a question like this is to determine which it’s asking for—permutations or combinations.  There is a clear difference between the two.

• Permutations are different orders of individual elements selected from a pool that is usually larger.  Order matters here, so just moving the same elements around creates a new permutation.
• Combinations, on the other hand, are different groups of individual elements selected from a pool that is usually larger. Order is irrelevant here, so simply rearranging the same elements doesn’t make a difference.

For example, say the TV show, The X-Factor, has three singers competing – Jack, Joe and Mary – with only two singing on a given day. The running order could see Jack sing first followed by Joe, or it could see Joe followed by Mary, or may be Mary will sing first.  There are six different permutations that the running order could take:

• Jack then Mary
• Jack then Joe
• Mary then Jack
• Mary then Joe
• Joe then Jack
• Joe then Mary

Combinations ignore order, so there are only three different possible combinations of singers for the day:

• Jack and Joe
• Jack and Mary
• Joe and Mary

Returning to our math problem, it asks how many teams of 5 players can be selected from 9 players who try out, without any particular order that the players are put into, so we are looking for the number of combinations, not permutations.

The Combination Formula

Now that we know we’re looking for the number of combinations, all we need to do is plug the numbers of our problem into a handy-dandy combination formula and watch the magic.  Let C be the number of combinations, n the number of elements in the larger pool, and r the number of elements selected.  The number of possible combinations is given by

n!
nCr = ____________

r!  (n – r)!

where ! means “factorial,” and works like this: 5! = 5x4x3x2x1, 4! = 4x3x2x1, and so forth.  We’re told that the total number of students on the team (n) is 9 and that the number selected (r) is 5, so plugging in, we have

9!                9!                    9 x 8 x 7 x 6 x 5!
9C5 =      ____________=    ___________=     __________________

5!  (9 – 5)!         5!  4!                5! (4x3x2x1)

Cancelling, we get

2
9 x 8 x 7 x 6 x 5!                 9 x 2 x 7
=           ___________________       =       ________________      =    9 x 2 x 7        =     126 combinations

5! (4 x 3 x 2)                          1

That’s it!

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Plan B: Process of Elimination

Multiple-choice questions dominate standardized tests, so it’s important for you to be able to quickly identify a correct answer choice in the midst of wrong ones. In some problems, this can be difficult, but have no fear—there’s a backup strategy you can use, known as Process of Elimination (POE).

What is Process of Elimination?

It’s not a strategy exclusive to solving exam questions. The world’s most famous detective, Sherlock Holmes, firmly believed in it and famously stated:

“When you have eliminated the impossible, whatever remains, however improbable, must be the truth.”

It’s about discarding the choices that are obviously wrong so you can focus on the remaining ones. If you’re having a hard time finding the right answer directly, eliminate the wrong ones first. What’s left must be correct, or even if you have more than one answer choice remaining, your odds of guessing the right one will have improved—1 out of 2 (50%) is much better than 1 out of 5 (20%).

When should it be used? How valuable a strategy is it when trying to get a high score?

If you know your Holmes, you know he used POE only when it was really necessary—the obvious was nowhere to be seen, and some serious assessment had to be given to a case before he could figure out whom the culprit was. On a standardized test, if you were to use POE for every question, you’d have to evaluate every single answer choice of every problem on the entire exam, exhausting yourself and probably running out of time in the process. You don’t need to use POE when the answer to a question is obvious—just choose it and move on.

Process of Elimination in Action

Let’s look at a brief example so you can see how best to apply POE. Read the sentence and associated question below, and check out the 5 answer choices.

Trevor was impressed by the enthusiasm his students showed when they were in class, and believed it was a good sign.

Q: The author of the text describes Trevor as being

1. downbeat about his students’ prospects in the run-up to an exam.
2. confident that all of his students will pass their exam.
3. angry that his students are wasting their time and his.
4. optimistic about his students’ chances because of their attitude.
5. all of the above

From the sentence, we expect the answer to be along the lines of Trevor believing something good about his students. When you look at the selection of answer choices, if the correct one is not immediately clear, evaluate each and eliminate those that cannot be correct.

A and C can be ruled out because the author states that Trevor is “impressed by the enthusiasm” being shown and saw it as “a good sign.” These are all positive details, so he was neither “downbeat” nor “angry,” which are negative.

E can be ruled out because some of the answer choices contradict each other, so the answer cannot possibly be “all of the above.”

Through POE, all three of these answer choices can be ignored.

1. downbeat about his students’ prospects in the run-up to an exam.
2. confident that all of his students will pass their exam.
3. angry that his students are wasting their time and his.
4. optimistic about his students’ chances because of their attitude.
5. all of the above.

The choice now comes down to B or D. Both have positive tones, which makes them much more plausible choices because they correspond to the positive tone of the sentence. However, if you read the original sentence carefully again, you’ll see there are informational differences which can help us to identify which is correct.

B states that Trevor is “confident that all of his students will pass their exam,” while D describes him as being “optimistic about his students’ chances.”

The original sentence mentions nothing about passing a test, so answer choice B can be eliminated too. The answer must therefore be D, which makes sense, because the sentence simply says that Trevor believed the enthusiasm of his students was “a good sign,” and D points to such optimism.

A Word of Caution!

Remember, it’s faster and requires less effort to simply pick out the right answer choice and move on this should always be your Plan A. Only if you’re having a hard time doing that, use Process of Elimination as Plan B.

Summer Test Prep Strategies

In our article, “Why Study For The GRE This Summer,” we discussed why summer is the ideal time for test prep. Now we’ll show you how to make the most of it and sufficiently prepare yourself for an upcoming standardized test.

Have a Study Plan to Prevent Procrastinating

We’ve all been guilty of avoiding a project or homework assignment, but students who procrastinate preparing for a standardized test often find themselves in a time crunch later on as test day and admissions deadlines approach. Creating a long-term study plan now can help ensure that you have sufficient time to prepare.