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605-655 (Medium)|   Combinations|                                 
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A researcher plans to identify each participant in a certain medical 17 Sep 2020, 20:25
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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

Say there are minimum of \(n\) letters needed, then;

The # of single letter codes possible would be \(n\) itself;
The # of pair of distinct letters codes possible would be \(C^2_n\) (in alphabetical order);

We want \(C^2_n+n\geq{12}\) --> \(\frac{n(n-1)}{2}+n\geq{12}\) --> \(n(n-1)+2n\geq{24}\) --> \(n(n+1)\geq{24}\) --> \(n_{min}=5\).

Answer: B.

Hope it's clear.
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Hi Bunuel, I am a confused by the answer. The question stem specifically mentions that the 2-letter codes need to be alphabetized. Doesn't this mean that order *does* matter in this question, so it's a permutation, not a combination.

So instead of \(C^2_n\), wouldn't it be \(P^2_n\) --> \(n(n-1)\). We only divide by \(2!\) when we are dealing with combinations where order doesn't matter, but with permutations where order does matter, the formula is different.

Can you explain the logic here? Thanks.
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A researcher plans to identify each participant in a certain medical 17 Sep 2020, 21:01
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Bunuel
sarb
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

Say there are minimum of \(n\) letters needed, then;

The # of single letter codes possible would be \(n\) itself;
The # of pair of distinct letters codes possible would be \(C^2_n\) (in alphabetical order);

We want \(C^2_n+n\geq{12}\) --> \(\frac{n(n-1)}{2}+n\geq{12}\) --> \(n(n-1)+2n\geq{24}\) --> \(n(n+1)\geq{24}\) --> \(n_{min}=5\).

Answer: B.

Hope it's clear.

Hi Bunuel, I am a confused by the answer. The question stem specifically mentions that the 2-letter codes need to be alphabetized. Doesn't this mean that order *does* matter in this question, so it's a permutation, not a combination.

So instead of \(C^2_n\), wouldn't it be \(P^2_n\) --> \(n(n-1)\). We only divide by \(2!\) when we are dealing with combinations where order doesn't matter, but with permutations where order does matter, the formula is different.

Can you explain the logic here? Thanks.
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Hi rvshchwl,

To start, this question can be solved rather easily with a 'brute force' approach (meaning that you don't need any special formulas, you just need to 'map out' the possibilities on your pad until you have the answer to the question that is asked - and several of the posts in this thread showcase that approach). If you do choose to approach this question with a formula, then it might help to think of a simple example first; that way, you can select the formula that fits the same logic.

The prompt tells us that any 2-letter codes MUST be written in alphabetical order. For example, if we had just two letters (A and B), then we could form just ONE code:

AB

By extension, with every pair of letters, we have just ONE code. Mathematically speaking, that's a Combination, not a Permutation (with 2 different letters and a Permutation, we would have 2 codes - but again, that's not allowed here).

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A researcher plans to identify each participant in a certain medical 21 Oct 2020, 14:01
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BrentGMATPrepNow
sarb
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

One approach is to add a BLANK to the letters in order to account for the possibility of using just one letter for a code.

ASIDE: Notice that, if we select 2 characters, there's only 1 possible code that can be created. The reason for this is that the 2 characters must be in ALPHABETICAL order. Or, in the case that a letter and a blank are selected, there's only one possible code as well.

Now we'll test the answer choices.

Answer choice A (4 letters)
Let the letters be A, B, C, D
We'll add a "-" to represent a BLANK.
So, we must choose 2 characters from {A, B, C, D, -}
In how many ways can we select 2 characters?
We can use combinations to answer this. There are 5 characters, and we must select 2. This can be accomplished in 5C2 ways (= 10 ways).
So, there are only 10 possible codes if we use 4 letters. We want at least 12 codes.

[i]ASIDE: If anyone is interested, we have a free video on calculating combinations (like 5C2) in your head below.

Answer choice B (5 letters)
Let the letters be A, B, C, D, E
Once again, we'll add a "-" to represent a BLANK.
So, we must choose 2 characters from {A, B, C, D, E, -}
There are 6 characters, and we must select 2. This can be accomplished in 6C2 ways (= 15 ways...PERFECT).

So, the least number of characters needed is 5

Answer: B
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Excuse me, but the problem says "alphabetical order".
As I get it "alphabetical" means:
AB
BC
CD
DE
etc., where all letters come in an alphabetical sequence.

AB - is an "alphabetical order"
AC, AD, AE, etc - is not very "alphabetical". There are not so many alphabets with such an order of letters.

Very unclear problem imho.
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A researcher plans to identify each participant in a certain medical 21 Oct 2020, 15:50
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Hi Antharas,

From what you mention in your post, I think that you might be confusing "alphabetical order" with "consecutive letters of the alphabet in order."

For example, with the letters A, B, C, D, E, there are a variety of pairs of letters that you could put in alphabetical order:
AB, AC, AD, AE, BC, BD, BE, CD, CE and DE

However, there are only 4 pairs that are consecutive letters in order:
AB, BC, CD and DE

Since the question is written in English, it's convenient to use the English alphabet as a reference, but you could use ANY alphabet (since there will be an 'alphabetical order' to it regardless of the country of origin).

GMAT questions are always carefully written, so you have to be mindful not to accidentally add any 'restrictions' to the information that prompt gives you.

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A researcher plans to identify each participant in a certain medical 26 Jan 2021, 18:18
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Hey,

I have a question regarding this. I decided not to use the regular formula and do it by trying. So my codes would be:

ZZ
YY
YZ
XX
XY
XZ
WW
WX
WY
WZ
Z
Y

I got 12 codes, all different, with only four letters. Can you please tell me what am I misinterpreting?
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A researcher plans to identify each participant in a certain medical 26 Jan 2021, 21:03
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madalenafnunes
Hey,

I have a question regarding this. I decided not to use the regular formula and do it by trying. So my codes would be:

ZZ
YY
YZ
XX
XY
XZ
WW
WX
WY
WZ
Z
Y

I got 12 codes, all different, with only four letters. Can you please tell me what am I misinterpreting?
Show more

Hi madalenafnunes,

The prompt tells us that the potential codes can either be a single letter or two DISTINCT letters written in Alphabetical order. This means that the letters must be DIFFERENT (for example, XX is NOT an option) and 'order' (for example, XY is an option but YX is NOT). Your list includes 4 codes of duplicate-letters that are not allowed.

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A researcher plans to identify each participant in a certain medical 26 Apr 2021, 20:53
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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
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Bunuel , VeritasKarishma , does the word pair have no significance here? i.e. can we even take a combination like ABC,BCD,CDE? Please advise.
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A researcher plans to identify each participant in a certain medical 26 Apr 2021, 23:34
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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

Bunuel , VeritasKarishma , does the word pair have no significance here? i.e. can we even take a combination like ABC,BCD,CDE? Please advise.
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No, you cannot take ABC as a code.

We are given that the code should be either a single letter so A or B or C etc or a pair of distinct letters written in alphabetical order so AB, AC, BC are acceptable codes but BA, ABC, AA are not.

So the question is that we should be able to make at least 12 total codes with the letters we pick.
If we pick 4 letters (A, B, C, D) , we can make 4 (single letter codes) + 4C2 (two letter codes) = 10 codes only.
Note that 4C2 gives you the number of codes with 2 letters. You do not need to arrange them because once you pick the letters, only one arrangement is possible since the letters must be placed alphabetically. So if I pick B and D, the only two letter code I get is BD.

If we pick 5 letters, we can make 5 + 5C2 = 15 codes.

Hence 5 letters are enough.
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A researcher plans to identify each participant in a certain medical 27 Apr 2021, 20:14
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Jitu20
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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

Bunuel , VeritasKarishma , does the word pair have no significance here? i.e. can we even take a combination like ABC,BCD,CDE? Please advise.
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Hi Jitu20,

Yes - the word "pair" is significant because it defines the limitations on which codes are possible. The prompt mentions two options (separated by the word "or" in the sentence):

A code can consist of:
1) a single letter (such as A or B or C, etc.)
2) a pair (meaning 'two') of distinct (meaning different) letters written in alphabetical order (such as AB or AC or AD or BC, but NOT BA nor CA nor DB, etc.).

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A researcher plans to identify each participant in a certain medical 29 Jul 2021, 01:27
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If it was mentioned in the question that the alphabetically placed codes had to be consecutive, then the answer would be (D) 7 right? This is exactly what I assumed when solving first and got it wrong.
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A researcher plans to identify each participant in a certain medical 29 Jul 2021, 16:54
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If it was mentioned in the question that the alphabetically placed codes had to be consecutive, then the answer would be (D) 7 right? This is exactly what I assumed when solving first and got it wrong.
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Hi LamboWalker,

The word "consecutive" does NOT appear in the original prompt. From what you mention in your post, I think that you might be confusing "alphabetical order" with "consecutive letters of the alphabet in order."

As an example, with the letters A, B, C, D, and E, there are a number of different pairs of letters that you could put in alphabetical order:
AB, AC, AD, AE, BC, BD, BE, CD, CE and DE.

In contrast, there are only 4 pairs that are consecutive letters in order (but this is NOT what the prompt described):
AB, BC, CD and DE

Since the question is written in English, it's convenient to use the English alphabet as a reference, but you could use ANY alphabet (since there will be an 'alphabetical order' to it regardless of the country of origin).

GMAT questions are always carefully written, so you have to be mindful not to accidentally add any 'restrictions' to the information that the prompt gives you.

GMAT assassins aren't born, they're made,
Rich
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A researcher plans to identify each participant in a certain medical 13 May 2022, 11:37
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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

Say there are minimum of \(n\) letters needed, then;

The # of single letter codes possible would be \(n\) itself;
The # of pair of distinct letters codes possible would be \(C^2_n\) (in alphabetical order);

We want \(C^2_n+n\geq{12}\) --> \(\frac{n(n-1)}{2}+n\geq{12}\) --> \(n(n-1)+2n\geq{24}\) --> \(n(n+1)\geq{24}\) --> \(n_{min}=5\).

Answer: B.

Hope it's clear.
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Kindly explain - why nc2 + n. I could not understand why you added the n.
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Kindly explain - why nc2 + n. I could not understand why you added the n.
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I LOVE permutations and combinations. I studied combinatorics and graph theory in college. If this were a forum about those topics, I'd gladly catch you up. But this is a forum about the GMAT, so let's not waste your time. Twenty years ago, GMAC drastically reduced the weight that it applies to combinations and permutations. They are barely tested nowadays, and when they are, it's not this cumbersome. If you find an OFFICIAL combinations or permutations question from the past ten years that you have trouble with, tag me and I'll help you out. But only if there isn't a super-easy way to avoid the "real math" and still get the question right.

38% of people have missed this question on this site. THIRTY-EIGHT!!!! On a question that a ten-year-old could get right just by listing the options! Why are we making things more complicated than they need to be? Do we get bonus points for knowing the "real math?" No. Get the question right and move on.

A
B
C
D
AB
AC
AD
BC
BD
CD

That's not enough. Add E and we can tell it will be enough without even listing any more options. Done.
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Seeking Help

I am curious why the order of the same letters cannot be reversed? For example AB and BA should considered as two pairs right? So it would be 2P4 instead of 2C4

Could anyone help me out?
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A researcher plans to identify each participant in a certain medical 16 May 2022, 08:07
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603987966
Seeking Help

I am curious why the order of the same letters cannot be reversed? For example AB and BA should considered as two pairs right? So it would be 2P4 instead of 2C4

Could anyone help me out?
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The red section of the question stem requires the letter to be in alphabetical order. BA is not an option.

sarb
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
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A researcher plans to identify each participant in a certain medical 27 Feb 2024, 18:28
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### Understanding the Problem
You need to find the smallest set of letters that can create unique codes for 12 participants. Codes can be a single letter or a pair of distinct letters in alphabetical order.

### Strategy
1. **Single Letters**: Each single letter can be a code by itself. So, if you have \(N\) letters, you get \(N\) codes right off the bat.
2. **Pairs of Distinct Letters**: If you have \(N\) letters, how many ways can you pair them up, ensuring they are distinct and in alphabetical order? This is a classic combination problem, where order doesn't matter, and you can't repeat letters. The formula for combinations is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n\) is the total number of items (letters, in this case), and \(k\) is the number of items to choose (2 for pairs).

### Goal
Find the smallest \(N\) such that the total number of codes (\(N + \text{combinations of N taken 2 at a time}\)) is at least 12.

### Calculation
1. **Single Letters**: \(N\) codes.
2. **Pairs of Letters**: \(\binom{N}{2}\) codes.

You need \(N + \binom{N}{2} \geq 12\).

Let's start with the smallest possibilities and work our way up:

- **For \(N = 4\)**: You get 4 single-letter codes. For pairs, \(\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6\) pairs. Total = \(4 + 6 = 10\) codes. That's not enough for 12 participants.
- **For \(N = 5\)**: You get 5 single-letter codes. For pairs, \(\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10\) pairs. Total = \(5 + 10 = 15\) codes.

### Answer
With \(N = 5\), you get 15 unique codes, which is more than enough for 12 participants, making it the least number of letters needed. So, the answer is:

B. 5
 
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A researcher plans to identify each participant in a certain medical 17 Jun 2024, 20:24
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Hi Bunnel

won't this \(C^2_n\)
just give you all the pairs available?
we need them also ordered....
Notice that we are told that letters in the code should be written in alphabetical order. Now, 2Cn gives different pairs of 2 letters possible out of n letters, but since codes should be written in one particular order (alphabetical), then for each pair there will be only one ordering possible, thus the number of codes out of n letters equals to number of pairs out of n letters.

Hope it's clear.
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­
I m getting confused regarding when to use Permutation and Combination. Here since we had a restriction of alphabetical order, was that the reason we used combinations? (Since it will be give unique pairs irrespective of the order)

Also, if the restriction of alphabetical order was not there, then could we use permutation i.e. nP2 in that case?

Moreover, can you please help me with some sample questions which can help me concretise this distinction?­
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Bunuel

ronr34
Hi Bunnel

won't this \(C^2_n\)
just give you all the pairs available?
we need them also ordered....
Notice that we are told that letters in the code should be written in alphabetical order. Now, 2Cn gives different pairs of 2 letters possible out of n letters, but since codes should be written in one particular order (alphabetical), then for each pair there will be only one ordering possible, thus the number of codes out of n letters equals to number of pairs out of n letters.

Hope it's clear.
­
I m getting confused regarding when to use Permutation and Combination. Here since we had a restriction of alphabetical order, was that the reason we used combinations? (Since it will be give unique pairs irrespective of the order)

Also, if the restriction of alphabetical order was not there, then could we use permutation i.e. nP2 in that case?

Moreover, can you please help me with some sample questions which can help me concretise this distinction?­
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We are interested in pairs of distinct letters written in alphabetical order; therefore, only one of the two possible orders is acceptable. That's why we use combinations (C) and not permutations (P).

­

21. Combinatorics/Counting Methods



For more:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.­
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gmathack93
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A researcher plans to identify each participant in a certain medical 06 Aug 2024, 11:39
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Bunuel

ronr34
Hi Bunnel

won't this \(C^2_n\)
just give you all the pairs available?
we need them also ordered....
Notice that we are told that letters in the code should be written in alphabetical order. Now, 2Cn gives different pairs of 2 letters possible out of n letters, but since codes should be written in one particular order (alphabetical), then for each pair there will be only one ordering possible, thus the number of codes out of n letters equals to number of pairs out of n letters.

Hope it's clear.
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­Hi Bunuel,

Thanks for explaining this, i think I am still not clear with this one. I too felt nPr should have been used here sinc order matters. However, as you point above (portion in the bold above), do you reckon if there is more than one ordering possible then we would have used nPr instead of nCr ??

Thanks in advance !!
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lnyngayan
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A researcher plans to identify each participant in a certain medical 04 Sep 2024, 02:37
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Bunuel
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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

Say there are minimum of \(n\) letters needed, then;

    The number of single-letter codes possible would be \(n\) itself;
    The number of two-letter codes possible (pair of distinct letters) would be \(C^2_n\) (in alphabetical order).

We want \(C^2_n+n\geq{12}\):

    \(\frac{n(n-1)}{2}+n\geq{12}\);

    \(n(n-1)+2n\geq{24}\);

    \(n(n+1)\geq{24}\);

    \(n_{min}=5\).

Answer: B.

Or else one could just test numbers.

    Four letters are enough for \(4+C^2_4=4+6=10\) codes. Not enough.
    Five letters are enough for \(5+C^2_5=5+10=15\) codes. Good!

Hope it's clear.

Hope it's clear.
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Could you explain why

\(C^2_n+n\geq{12}\) becomes \(\frac{n(n-1)}{2}+n\geq{12}\)?
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