12 Days of Christmas GMAT Competition - Day 8: A certain five-letter

1. he code consists of five letters: x1,x2,x3,x4,x5 where each letter is either A,B or C.
2. If A appears in any position but the first, it must be immediately preceded by B.
3. If B appears in any position but the first, it must be immediately preceded by C.

We need to find x5.

Statement (1):
The first and fourth letters in the code are the same, as are the second and fifth letters.

As per the given stmt, We can get both of these combinations B A C B A, A C B A C & C B A C B.

Hence x5 is not consistent. So AD can be eliminated.

Statement (2):
There is exactly one C in the five-letter code.

As per given conditions CBACBACBACBA.... You can pick any 5 from this series to satisfy the initial conditions. Using stmt 2 the only
combination that is possible is.

B A C B A. Stmt 2 alone is enough

Hence IMO B

Given (1) we could fill the code as A,C,B,A,C or B,A,C,B,A. Not sufficient
Given (2) we can only fill the code as B,A,C,B,A, Sufficient. Answer B

Hi everyone

We have 3 letters: A, B, C
We have 5 letter code: __,__,__,__,__
Given 2 constraints: if A->B , B , A , , B->C , C , B , ,
the first letter doesn't have any constraint.

[?]: what is the 5th letter?

(1) X , Y , , X , Y
Case 1: B , A , C , B , A - Letter A
Case 2: C , B , A , C , B - Letter B
Case 2: A , C , B , A , C - Letter C
Clearly - Insufficient

(2) only 1 C
Case 1: B , A , C , B , A
there is no other combination possible that has only 1 letter "C".
Sufficient.

Answer B

(1) 1ST = 4TH AND 2ND = 5TH
+ Previous constraints: BA OR CB needed, if not in the first positions thus let's try

1: A, 2:C, 3:B, 4:A, 5:C
OR
1:B, 2:A, 3:C, 4:B, 5:A

not sufficient. ELIMINATE A,D

(2) only one C + previous constraints: BA or CB needed, if not in the first positions, let's find a value
1:C, 2:B, 3:A, 4: not B not A not C... not possible

1: A, 2:not a, not b, C; 3:B; 4: A; not a, not b, not another c.... not possible

1:B, 2:A, 3:C, 4:B, 5:A only possible solution. Sufficient. Answer B

(1)
BACBA, CBACB and ACBAC match the rules and give us different answers.

Condition (1) is insufficient

(2)
The unique possible solution is BACBA.

Condition (2) is sufficient

Answer B

We are given a five-letter code with A, B, and C.

Given: A must be preceded by B unless it’s first, and B must be preceded by C unless it’s first. We need to find the fifth letter.

Statement 1 says the first and fourth letters are the same, and the second and fifth letters are the same. Possibilities:

1. B, A, C, B, A. - > A is the fifth letter
2. C, B, A, C, B. - > B is the fifth letter

No unique answer

Statement 1 is not sufficient

Statement 2 says there is exactly one C.

Because of the constraints of A and B, the sequence can only be B, A, C, B, A. - > A is the fifth letter

Statement 2 is sufficient

Answer is B.

1 2 3 4 5

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.

A B C A B & B C A B C; both satisfy the above-mentioned condition; hence, this is not sufficient alone.

(2) There is exactly one C in the five-letter code.

A B C A B & A A B C A; both satisfy the above-mentioned condition; hence, this is not sufficient alone.

But when both conditions are taken together, there would be one unique answer A B C A B; Both together are sufficient.

From the constraints we get that,
B A (if A isn't in the first position)
C B (if C isn't in the first position)

i) We can get a lot of cases like,
CBACB; BACBA; ACBAC - Hence, INSUFFICIENT

ii) If C only comes 1 time, the only possible code is
BACBA
5th letter - A

SUFFICIENT

IMO B

We are tasked to determine the fifth letter in the code. Let’s analyze the problem step by step.

Problem Setup:

1. The code consists of five letters: L1,L2,L3,L4,L5


2. The letters can only be A, B, or C.


3. The following constraints apply:

If an A appears anywhere except the first position, it must be immediately preceded by a B.

If a B appears anywhere except the first position, it must be immediately preceded by a C .




Information from the Statements:

Statement (1):

"The first and fourth letters in the code are the same, as are the second and fifth letters."

Denote L1=L4 and L2=L5.

This narrows the possible values of the code because certain patterns are forced to repeat.


Statement (2):

"There is exactly one C in the five-letter code."

This tells us there is exactly one C, and the remaining letters must be A and B.

Combining this with the rules for A and B , we can deduce specific sequences.



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Evaluate Sufficiency of the Statements:

Consider Statement (1) Alone:

1. If L1-L4 and L2=L5, the code structure becomes:



L1, L2, L3, L1, L2

3. However, no information is given about whether C appears in the code or how the specific rules for A and B interact. Thus, there are multiple possibilities:

For example, L1=A,L2=B & L3=C gives A,B,C,A,B, but other arrangements (like B,C,A,B,C ) are also possible.

Hence Statement (1) alone is not sufficient & we can thus eliminate options (A) & (D)


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Consider Statement (2) Alone:

1. With exactly one C in the code:

C can appear in any position L1,L2,L3,L4 or L5 .

The constraints on A and B must still hold.



2. Without additional information about the structure or positions of L1 and L2, there are still multiple valid arrangements.



Statement (2) alone is not sufficient. Hence option (B) can be eliminated.


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Combine Statements (1) and (2):

1. From Statement (1), L1-L4 and L2=L5, so the structure is:



L1, L2, L3, L1, L2

3. Since A and B are restricted by the rules:

A (if not first) must follow B .

B (if not first) must follow C .



4. Place C in L3, as it cannot repeat and does not violate constraints:



L1 = A, L2 = B, L3 = C, L4 = A, L5 = B

The fifth letter L5 is B.


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Final Answer:

Hence Both statements together are sufficient & hence the answer is option (C)

Solve this step by step:

1. Basic rules:
   
   • A must be preceded by B (except in the first position)
   • B must be preceded by C (except in the first position)
2. Statement (1):
   
   • Letters 1 and 4 are the same
   • Letters 2 and 5 are the same
   • Can't determine 5th letter just from this
3. Statement (2):
   
   • Exactly one C in the code
   • Think about where B can be:
     
     • B needs C before it (except the first position)
     • Since only one C can have at most one B after position 1
     • Similarly, at most one A after any B
   • Still multiple possibilities
4. Statements (1) + (2):
   
   • First and fourth letters same
   • Second and fifth letters same
   • One C
   • Due to rules about B needing C and A needing B:
     
     • The C must be followed by B
     • That B must be followed by A
   • This means the second letter (and thus the fifth letter) must be A

Answer: C

1: says A,B,C,A,B. it Satisfies
2: says one C is present but doesn't fully determine 5th letter
Therefore, statement 1 alone is sufficient

Hi All,

given that C comes ahead of B if B is not at first position and B comes ahead of A if A not a first position.

This constricts the that C will be at the first position therefore current positioning C B A _ _


Statement 1:

from this statement we can deduce C B A C B,

therefore we know B is the 5th digit.

Sufficient


Statement 2:

if we follow this statement then for the 4th position it will be wither A or B, but if A is the 4th digit then according to the premise B should be ahead therefore code cannot be formed

if we out B is the 4th digit but this will also invlaidate the premise as if B is not at the 1st place then C should be ahead and C cannot be ahead because C is already at the first place

In-sufficient


Therefore Ans A

There are 3 possibilities given the conditions: ACBAC, BACBA, CBACB [starting with each A,B,C]
clearly the statement 1 doesn't give any additional information
using Statement 2 we have BACBA ==> A is the fifth letter conclusively
Hence Statement 2 alone sufficient.

I would say answer C

We know that somewhere in the code, we have ABC (in this order)

(1) tell us that Digit 1 = digit 4 and digit 2 = digit 5
It remains several solutions

(2) allow us to choose between these two solutions

Answer is B. Statement 2 alone is sufficient

1) There are three possible codes: ACBAC, BACBA, or CBACB. This is insufficient.
2) There is only one possible code: BACBA. This is sufficient.

Let the 5 letter-code be XYZUV. What is V?

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
We don't know V. The code can be CCCCC or CBACB, etc.

(2) There is exactly one C in the five-letter code.
We know that CBA must go together. That means B must be right after C, and A must be right after B. So C must be the only letter to appear after A. Therefore, if A is not the last letter, then C must go after A.

In order to have exactly 1 C, we need to have A as the last letter. That means XYZUV = XYCBA. The 2nd letter can't be another C, and also can't be B (requiring the 1st letter to be C). So the first 2 letters must be BA.

So from (2), we know that the code must be BACBA.

Answer: B

P.S. This is a really hard question. I wonder if anyone can make it within 5 mins

Let's see all the positions A and B can take:

a
b	a
c	b	a
	c	b	a
		c	b	a

From here we can see that there're quite a lot of gaps to fill; if they're filled with a's or b's, we need to also add c's. So, if we don't have room for the full CBA set-up or even partial CB set-up, the only way to fill solitary cells is with a C, as we don't have any conditions for it.

(1) Let's fill in the added letters (1=4 and 2=5), and also add the necessary b-c elements as per conditions:

a	c	b	a	c
b	a	c	b	a
c	b	a	c	b
a	c	b	a	c
b	a	c	b	a

As we see in column 5, there're different possible answers. Insufficient.

(2) As we have a lot of gaps in the first set-up, let's check all the situations with only one C:

b	a	c	b	a
b	a	c	b	a

As a result, we see that there's only one option for A.
It's sufficient, so the answer is B.