How to Become a Great GMAT Guesser - Part 3
Back by popular demand! Lets talk some more about how to guess on the GMAT.
If you havent yet seen the earlier installments, I recommend starting at the beginning.
Try out this problem from the free GMATPrep tests. (If you know how to do the math for real, pretend you dont have enough time to do it. The test is about to end! How would you guess?)
*A company plans to assign identification numbers to its employees. Each number is to consist of four different digits from 0 to 9 inclusive, except that the first digit cannot be 0. How many different identification numbers are possible?(A) 3,024
(B) 4,536
(C) 5,040
(D) 9,000
(E) 10,000
So, what did you guess? And why did you guess that?
First, glance at the answers. Notice anything?
Theyre spread pretty far apart. When the answers are this far apart, there may be a chance to estimate to eliminate some answer choices.
Lets see if we can figure out whats going on.
Four digit number different digits cant start with 0 hmm. I might jot down something like this on my scrap paper:
___ ___ ___ ___
And then Id start playing with some numbers to understand whats going on. If there were no restrictions at all, then the possible numbers would range from 0000 to 9,999, for a total of 10,000 different numbers.
There are restrictions, though, so (E) cant be the correct answer. Are there enough restrictions to knock the total number down below 9,000?
The number cant start with a 0, so all the numbers from 0000 to 0999 are invalid. Thats 1,000 numbers, and were down to 9,000.
There are more restrictions from here (no repeated numbers), so (D) cant be the answer either. We just got rid of two answers not because we know how to answer this thing, but because we know how not to answer it. Thats a key skill if you want a good score on the GMAT!
From here, its not so easy to continue with this method and knock out more answers. At this point, Id guess from among the remaining three and move on. The good news is that it might only take about 45 to 60 seconds to knock out (D) and (E), so I can make a good guess while still saving some time.
Why would I do that, instead of trying to solve this in the first place? First, I might simply be running out of time; the section might be about to end on me.
Alternatively, I might be behind on time in the section overall and know that I need to save some time somewhere. I might choose to do that on an area in which Im weaker, or on what looks to me like a hard problem or a time-waster (something that's going to take a lot of work).
In this case, combinatorics is definitely a weaker area for meI hate these! If theres a trap, Ill usually fall into it, and I make careless mistakes. Even if Im not behind on time, Id consider doing what we just did and moving on; I cant get super frustrated by a problem type that I hate if I dont bother to try to do it in the first place!
In fact, Id feel pretty good about being able to narrow to 3 while also saving time, since I know these drive me crazy. And thats a smart move psychologically.
(Just a note: if I were far enough behind on time, I wouldn't even try to make an educated guess; I'd pick my favorite letter and move on as soon as I saw it was combinatorics.)
Okay, do you want to know how to do this one for real? (Theres still an important lesson to learn here, even if you do know how to do the official math!)
Because a digit cant be repeated, theres a pretty straightforward technique you can use to solve. If you can figure out how many possible numbers there are for each blank, you can multiply those possibilities together to get the total number of permutations.
___ ___ ___ ___
Consider that first blank, which represents the first digit. How many different possible numbers can go in that blank?
There are ten digits from 0 to 9 (count them!), but the first blank cant be 0, so that leaves us with just nine possible digits: 1, 2, 3, 4, 5, 6, 7, 8, and 9. Put that number in the first blank and put parentheses around it or add a multiplication symbol after it.
(9) ___ ___ ___
(Note: this number represents the number of possibilities for that blank; it doesnt mean that were starting the 4-digit number with the number 9.)
Next, how many possibilities are there for the second blank? Remember, no repeats!
We have to remove one number from contention (whatever went in the first blank) but we can use the 0 now. For instance, pretend that the first digit was a 7. The possible numbers for the second blank are 0, 1, 2, 3, 4, 5, 6, 8, and 9.
So weve got 9 possible digits again for the second blank:
(9)(9) ___ ___
Now, for the third blank, we have to knock out two digits because we've used up two numbers for the first and second blanks. That leaves us with 8 remaining digits (out of the original 10). The process is the same for the fourth blank: there are 7 remaining digits there.
(9)(9)(8)(7)
Wow. Thats kind of annoying without a calculator, huh? I wonder whether theres a way to avoid doing all that math
Of course there is. :D First, simplify a little bit by doing some multiplication, but nothing too annoying:
(9)(9)(8)(7) = (81)(56)
Glance at your answers again. (Always keep your answers in mind while working on problem solving questions!)
Take a look at the units digits of (81)(56). They determine the units digit of the product. In this case, the units digit is (1)(6) = 6. Only one answer choice has 6 in the units digit!
The correct answer is (B).
Now, you might be wondering: but that wont always work! Three of the answers all have 0 as the units digit!
Thats true. But remember that two of those three answers can be eliminated via our estimation method from earlier. Sometimes, its worth it to do that kind of thinking even if you are going to do the actual math!
So if you thought from the beginning, No problem, I know how to do that! Its just (9)(9)(8)(7)! thats great, but dont think that you dont have anything to learn on this problem. If you know how to do that already, then you can also learn to estimate pretty quickly that 9,000 and 10,000 cant possibly be right.
Its also not just a fluke that the three remaining answers each have a different units digit (4, 6, and 0). The people who write these test questions are very smart. Theyve thought of everything. They are going to create these kinds of shortcuts for the people who know how to take advantage ... dont you want to be one of those people?
If you can learn to pick these problems apart, understand how theyre constructed, and use that to your advantage when solving, youll become a great GMAT test-taker too.
By the way, have you figured out yet where the two remaining wrong answers come from? If you forget about the constraint that the number cant start with 0, then youll come up with (10)(9)(8)(7) = 5,040. Alternatively, if you forget to add the 0 possibility back in for the second digit, then youll come up with (9)(8)(7)(6) = 3,024.
Dont forget to join us next week for the final installment in this series.
Key Takeaways to Become a Great Guesser
(1) Dont just study how to correctly answer problems using real math. Use estimation, logic, and common-math-sense to get rid of wrong answers. This problem is a classic case of I might not know the right way to do this, but I can still tell that (D) and (E) are the wrong way. This is a business test! Use every advantage you can find.
(2) Even if you can actually do the math from there, narrowing down the answers can still save you some annoying computations on the back end of the problemand that leaves you more time and mental energy to tackle even harder problems, thereby lifting your score!
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.