A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code? GMAT Explanation, Video Solution, and Additional Practice Questions!

A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

A. 4

B. 5

C. 6

D. 7

E. 8

We need to help this researcher identify 12 participants using single letters or pairs of letters in alphabetical order. Key info: we want the LEAST number of letters to make these 12 codes. The word code should trigger: ordering! Yes, codes usually involve an ordering element. However, not always and this is one of the exceptions. Whenever a code has to be in alphabetical order or ascending/descending order it is highly likely that it no longer has an ordering element and should be treated as a group.

So what does that mean for our researcher and his medial experiment? Well, let's start testing the answers to find the least value that can create 12 codes. With that in mind start with the smallest number. Why? Because if the smallest number works then it's the correct answer. If say, 6 works, then you still need to test 5 and potentially 4. If looking for a max you might start with the biggest answer choice.

Ok, so how do we test 4? Well, you have 4 single digit codes say A, B, C, D.

But how about the pairs? Again, think of this as a group. So it's a group of 2 with 4 choices. Or 4 choose 2.

4*3/1*2 = 6

So 4 (single) + 6 (pairs) = 10.

Not enough to identify our twelve participants!

Let's try 5.

A, B, C, D, E

5 single codes.

5*4/1*2 = 10

10 pairs

10 + 5 = 15. That works!

So the least number of letters needed to identify the 12 participants: 5. B.

That's one way to do it. There's also a very practical way to approach this combinatorics puzzle. And, all over the GMAT, especially on combinatorics questions with a very limited numbers of possibilities (we're only looking for 12 codes here) remember to stay down to earth. Sometimes you can just make a list. And sometimes that will be the fastest approach. Again: stay practical on the GMAT!

So let's try 4 letters.

Single Letter Codes: 4

A, B, C, D

Pairs: 6

AB. BC. DE

AC. BD

AD

Total Codes: 10

Let's try 5 letters.

Single Letter Codes: 5

A, B, C, D, E

Pairs: 10

AB  BC. CD. DE

AC. BD. CE

AD. BE

AE

Total Codes: 15

If you're taking a practical approach take your time planning. Think about how you want to make your list. Don't rush it. What I see a lot in GMAT tutoring is students having a decent understanding of how to approach things from a practical perspective BUT because they're not using a formula or test prep technique don't feel confident and then rush the setup/execution. Then they get the feeling that the practical approach doesn't work and that they need a formula for everything. So, again, if you're going for a non-formulaic setup, which sometimes is exactly what you should do, give it the space to succeed.

A researcher plans to identify each participant in a certain medical experiment GMAT Explanation Diagram

A researcher plans to identify each participant in a certain medical experiment GMAT Explanation 2 Diagram

Video Solution: A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

Additional GMAT Combinatorics Practice Question Examples

Here is a grouping question from the GMAT Prep Tests. This is using the same method of grouping as the question above (although it has different conditions/constraints).

Another grouping question but this time Data Sufficiency from GMAT Question of the Day

Here is a somewhat standard GMAT Combinatorics Code example from GMAT Question of the Day

Finally another wordy combinatorics example from question of the day

 

From a group of 8 volunteers, including Andrew and Karen, 4 people are to be selected at random to organize a charity event. What is the probability that Andrew will be among the 4 volunteers selected and Karen will not?

(A) 3/7
(B) 5/12
(C) 27/70
(D) 2/7
(E) 9/35

From a group of 8 volunteers, including Andrew and Karen, 4 people are to be selected at random to organize a charity event

Additional GMAT Combinatorics/Grouping Practice Questions

Here's another grouping question from the GMAT Official Guide: A researcher plans to identify each participant in an experiment

And another grouping question from GMAT Question of the day

A certain office supply store stocks 2 sizes of self-stick notepads, each in 4 colors: Blue, Green, Yellow or Pink. The store packs the notepads in packages that contain either 3 notepads of the same size and the same color or 3 notepads of the same size and of 3 different colors. If the order in which the colors are packed is not considered, how many different packages of the types described above are possible?

(A) 6
(B) 8
(C) 16
(D) 24
(E) 32

A certain office supply store stocks 2 sizes of self-stick notepads each in 4 colors Blue Green Yellow or Pink

 

A basket contains 5 apples, of which 1 is spoiled and the rest are good. If Henry is to select 2 apples from the basket simultaneously and at random, what is the possibility that the 2 apples selected will include the spoiled apple?

(A) 1/5
(B) 3/10
(C) 2/5
(D) 1/2
(E) 3/5

GMAT tutoring students tend to dislike probability questions. A couple of things to remember regarding GMAT probability:

  1. It doesn't come up that much.
  2. When it does come up often the question isn't that challenging.
  3. If you do get a challenging probability question you can always skip.

Let's focus on the question first: what is the possibility that the 2 apples selected will include the spoiled apple?

We need a spoiled apple selected. OK. There's a very practical way of doing this:

G1 G2 G3 G4 S

10 Possible pairs. 4 of them have a spoiled apple. So 4/10 or 2/5 chance of having a pair with the spoiled apple.

G1 G2

G1 G3

G1 G4

G1 S

G2 G3

G2 G4

G S

G3 G4

G3 S

G4 S

Easy. If the numbers are small there's nothing wrong with writing things out. In fact, it can be the best approach.

The other way to do it is with the slot method. Calculate the number of ways to create a group of two from 5 things. That will be your total or your denominator. And then calculate the number ways to create a group of two with the constraint that one of those things (the spoiled apple) must be included. That will be your numerator. Remember that probability is just specific scenario/all scenarios.

5*4/1*2 = 10 (the number of ways to create a pair from 5 things)

With one of the spots reserved for the spoiled apple you're only left with the other spot to populate. So how many things can go in that spot? 4 (the 4 good apples). So the numerator is 4 and the denominator is 10. 4/10 = 2/5.

If two of the four expressions x+y, x+5y, x-y, and 5x-y are chosen at random, what is the probability that their product will be of the form of x^2 - by^2, where b is an integer?

(A) 1/2

(B) 1/3

(C) 1/4

(D) 1/5

(E) 1/6

Be super comfortable with difference of squares. You're almost guaranteed to need it on your GMAT. You need to know it forwards, backwards, and upside-down. There isn't much content you need to know for the GMAT quant but in order to be consistently successful so that you can go in there on test day and snag your 700+ score you need complete fluency with that limited content. On this one we're looking for a product of two terms that will produce difference of squares. So we need something in the format: (x+y(x-y). There's only one pairing that gives us the right format. Now you have to figure out what that means in terms of probability. In general, probability is specific scenario/total scenarios. You can calculate the total and the specific scenarios using the slot method. How many teams of 2 can you make from four things? (4*3)/(1*2) = 6. So that's your denominator. And then there's only one team that works, the pairing of (x + y)(x - y). So that leaves you with 1/6. E. Detailed diagrams below and an in depth video explantation here: If two of the four expressions

 

If two of the four expressions x+y, x+5y, x-y, and 5x-y are chosen at random

You also can think about this in terms of straight probability and say: 2 choices our of 4 and then 1 choice out of 3. 2/4 * 1/3 = 2/12 = 1/6. Comment with any questions or additions. Happy studies!

If two of the four expressions x+y, x+5y, x-y, and 5x-y are chosen at random alt

 

 

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