PS - combination

Question

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

Brent@GMATPrepNow · Answer

abhasjha wrote: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.

ASIDE: If anyone is interested, we have a free video on calculating combinations (like 5C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

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

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

Brent@GMATPrepNow · Answer

abhasjha wrote: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 can also TEST each answer choice by LISTING all possible codes.

Answer choice A (4 letters)
Let the letters be A, B, C, D
The possible codes are:
A
B
C
D
AB
AC
AD
BC
BD
CD
TOTAL = 10 (not enough. We need at least 12 codes)

Answer choice B (5 letters)
Let the letters be A, B, C, D, E
The possible codes are:
A
B
C
D
E
AB
AC
AD
AE
BC
BD
BE
CD
CE
DC
TOTAL = 15

Perfect, 5 letters will give us the 12 codes we need.

Answer: B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

Mathsbuddy · Answer

Imagine a 2 digit number AB where A and B are integers >= 0
Let's see what is the lowest base that can generate the value twelve using just these 2 digits?
Using base 4, the maximum number achievable using 2 digits is 33 (base 4) = 3x4 + 3 = 15 (denary)
However we need to remove 3 doubles (where A=B): 11,22,33 which leaves us with 15-3 = 12
and also remove half of these (as AB is accepted, but not BA), leaving us with 6
We can also add 4 for the 4 single digits (0,1,2,3)
Therefore Base 4 produces 10 possibilities
Hence any greater base will suffice (as the next base will produce plenty more)
Therefore base 5 is the lowest base.
Answer = B) 5.

Mathsbuddy · Answer

Mathsbuddy wrote:Imagine a 2 digit number AB where A and B are integers >= 0
Let's see what is the lowest base that can generate the value twelve using just these 2 digits?
Using base 4, the maximum number achievable using 2 digits is 33 (base 4) = 3x4 + 3 = 15 (denary)
However we need to remove 3 doubles (where A=B): 11,22,33 which leaves us with 15-3 = 12
and also remove half of these (as AB is accepted, but not BA), leaving us with 6
We can also add 4 for the 4 single digits (0,1,2,3)
Therefore Base 4 produces 10 possibilities
Hence any greater base will suffice (as the next base will produce plenty more)
Therefore base 5 is the lowest base.
Answer = B) 5.

Using the same algorithm for base 5, we get:
Maximum number = 44 (base 5) = 4 x 5 + 4 = 24 (denary)
Remove 4 doubles (11, 22, 33, 44) -> 24 - 4 = 20
Half this to remove alphabetical reversals -> 10
Add 5 for the 5 single digits -> 10 + 5 = 15
15 > 12, so base 5 works
Answer = B) 5

GMATinsight · Answer

abhasjha wrote: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 can also TEST each answer choice by LISTING all possible codes.

Answer choice A (4 letters)
Let the letters be A, B, C, D
The possible codes are:
A
B
C
D
AB
AC
AD
BC
BD
CD
TOTAL = 10 (not enough. We need at least 12 codes)

Answer choice B (5 letters)
Let the letters be A, B, C, D, E
The possible codes are:
A
B
C
D
E
AB
AC
AD
AE
BC
BD
BE
CD
CE
DC
TOTAL = 15

Perfect, 5 letters will give us the 12 codes we need.

Answer: B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

Gurpreet singh · Answer

Brute force 1 A 2 B 3 AB 4 C 5 CA 6 CB 7 D 8 AD 9 DB 10 DC 11 E 12 EA Alphabets used 5

Jeff@TargetTestPrep · Answer

abhasjha wrote: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 can also TEST each answer choice by LISTING all possible codes.

Answer choice A (4 letters)
Let the letters be A, B, C, D
The possible codes are:
A
B
C
D
AB
AC
AD
BC
BD
CD
TOTAL = 10 (not enough. We need at least 12 codes)

Answer choice B (5 letters)
Let the letters be A, B, C, D, E
The possible codes are:
A
B
C
D
E
AB
AC
AD
AE
BC
BD
BE
CD
CE
DC
TOTAL = 15

Perfect, 5 letters will give us the 12 codes we need.

Answer: B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

Rich.C@EMPOWERgmat.com · Answer

Hi All,

We’re told that each participant in an experiment will be referred to with a ‘code’ that is either 1 letter or 2 DISTINCT letters written in ALPHABETIC ORDER. We’re asked for the LEAST number of letters that would be needed to give each of 12 participants a unique code.

Since the answers are numbers, we can TEST THE ANSWERS solve this problem.

IF… there were 4 letters, then we can refer to those letters as A, B, C and D – and the possible codes would be:

A, B, C and D = 4 one-letter codes
AB, AC, AD, BC, BD, CD = 6 two-letter codes
Total = 10 codes… which is close, but too few.

Adding 1 additional letter will clearly create more than 12 codes, so you can stop working at this point. If you choose to list out the codes though, then we can refer to the letters as A, B, C, D and E – and the codes would be:

A, B, C, D and E = 5 one-letter codes
AB, AC, AD, AE, BC, BD, BE, CD, CE and DE = 10 two-letter codes
Total = 15 codes… which is more than we technically need, but the least number of total letters that will give us 12 total codes.

Final Answer: B

GMAT Assassins aren’t born, they’re made,
Rich

Contact Rich at [email protected]

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