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805+ (Hard)|   Combinations|         
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 09:00
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12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A certain five-letter code may only be constructed with A’s, B’s, and C’s. If an A appears in any position but the first, it must be immediately preceded by a B. If a B appears in any position but the first, it must be immediately preceded by a C. What is the fifth letter in the code?

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
(2) There is exactly one C in the five-letter code.


 


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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 10:00
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Bunuel
12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A certain five-letter code may only be constructed with A’s, B’s, and C’s. If an A appears in any position but the first, it must be immediately preceded by a B. If a B appears in any position but the first, it must be immediately preceded by a C. What is the fifth letter in the code?

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
(2) There is exactly one C in the five-letter code.


 


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Manhattan Prep Official Explanation:



This is a Data Sufficiency Logic problem. Carefully note the constraints: five letters; A’s, B’s, or C’s; any A not in the first position is immediately preceded by a B, and any B not in the first position is immediately preceded by an A. The question asks you to choose one (A, B, or C) for the fifth letter.

Statement (1): INSUFFICIENT. Multiple codes are possible with different fifth letters: ACBAC, BACBA, BCCBC, CBACB, CBCCB, CCBCC, CCCCC.

Statement (2): SUFFICIENT. If there is only one C in the code, then there must be a total of four A’s and B’s. There cannot be four A’s because a B would need to precede any A that did not appear in the first spot of the code. For a similar reason, there cannot be three A’s and only one B. There cannot be three or four B’s because the code would demand more than a single C. Thus there must be two A’s and two B’s.

As any B not in the first position must be preceded by a C, the only way there can be fewer C’s than B’s is for the code to begin with a B. Because there are equal numbers of A’s and B’s, and because B is the first letter of the code, A must be the second letter. C must be the third letter because it is the only letter that can be preceded by an A, leaving B and A as the fourth and fifth letters of the code, respectively: BACBA.

The correct answer is (B): Statement (2) alone is sufficient, but statement (1) is not sufficient.
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 09:45
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(1) There are many possibilities for a 5th letter - Insufficient
(2) Only the possibility ABCAB is valid, therefore the 5th letter is B - Sufficient

Option B
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 09:53
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Bunuel
12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A certain five-letter code may only be constructed with A’s, B’s, and C’s. If an A appears in any position but the first, it must be immediately preceded by a B. If a B appears in any position but the first, it must be immediately preceded by a C. What is the fifth letter in the code?

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
(2) There is exactly one C in the five-letter code.


 


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Case 1: not sufficient: There exist 2 cases: ACBAC and BACBA fifth letter can be C or A

Case 2: sufficient:

Start with A: second letter can't be A, bcz first shud be B, can't be B bcz C shud be first hence AC_ _ _
Third letter can be B fourth A = ACBA_ not possible to put any A,B,C => hence not possible

Similarly doing for B and C we would realize that only BACBA is the only case possible with exactly one C.

Hence B
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 09:56
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A certain five-letter code may only be constructed with A’s, B’s, and C’s. If an A appears in any position but the first, it must be immediately preceded by a B. If a B appears in any position but the first, it must be immediately preceded by a C. What is the fifth letter in the code?

Let the five-letter code be L1 L2 L3 L4 L5; where L1 is first letter, L2 is second letter and .... L5 is fifth letter.

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
L1=L4; L2=L5
If L2 = A = L5; L1=B = L4
5-letter code = BACBA
If L2 = B = L5; L1 = C = L4
5-letter code = CB(A/C)CB
If L2 = C = L5;
5-letter code = (A/B/C)CBAC
Fifth letter in the code may be any one of A, B or C.
NOT SUFFICIENT

(2) There is exactly one C in the five-letter code.
If C is first letter; CBA_ _ ; Not possible
If C is second letter: _CBA_; Not possible
If C is third letter: BACBA; Possible
If C is fourth letter: _BACB: Not Possible
If C is firth letter : _ _ BAC; Not possible
Only possible 5-letter code is BACBA
The fifth letter in the code = A
SUFFICIENT

IMO B
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 09:58
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If you start with all the possible cases you will be lost. In DS question the correct approach ALWAYS is to prove insufficiency
So basically I try to show that three are 2 possibles cases that follow the given conditions and hence statement is insufficient.

St 1) The first and fourth letters in the code are the same, as are the second and fifth letters.
Very easy to show two possible cases.

C C C C C
A C C B A

2 cases, Insufficient! MOVE ON.

St 2) There is exactly one C
Note: the conditions say for any other place than the first place if I use A , B must precede A { B A}
and for any other place than the first place if I use B , C must precede B { C B}

lets start with last position, and note "exactly' one C

1. Assume last place is C
- - - - C. for the 4th place I am left with either A or B.
- C B A C
- - C B C
You can see whatever case I use I will need one more C

2. Assume last place is A
Then I am sure that - - C B A will be the pattern
What can second place be? if I use B then I will need one more C, Clearly A is the second place and So first place should be B
B A C B A This is one possible case, is there any other possible case?

3. Assume last place is B
- - - C B This will be my starting case. What can 3rd place be?
if its A " - B A C B " Now the first place is forced to be C again, but remember we only had one C. not possible
if its B " - C B C B" second place is forced to be C again. not possible.

SO for Statement 2 i could only find one possible case Hence St 2 Alone Sufficient Option B
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 10:09
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Statement 1

There are two possible cases that satisfy all the conditions,

a. A C B A C and b. B A C B A

The fifth letter in the code can be either A or C. NOT SUFFICIENT

Statement 2

Let us check by placing one C at each position of _ _ _ _ _ and test whether multiple cases are possible

C B A _ _ we can't place A or B on the 4th position
B C B A _ we can't place A or B on the 5th position
B A C B A - A is the 5th digit
B A _ C _ we can't place A or B on the 3rd position
B A _ _ C we can't place A or B on the 3rd position

We have only one valid case. SUFFICIENT

Answer B.
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 10:10
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Bunuel
12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A certain five-letter code may only be constructed with A’s, B’s, and C’s. If an A appears in any position but the first, it must be immediately preceded by a B. If a B appears in any position but the first, it must be immediately preceded by a C. What is the fifth letter in the code?

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
(2) There is exactly one C in the five-letter code.


 


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for the 12 Days of Christmas Competition

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1)

We can have A C B A C or B A C B A. As we have two possible combinations, we can eliminate A, and D.

2)

Except when B is at the first position, it must be preceded by C. Hence, C B must appear together.

Except when A is at the first position it must be preceded by B. Hence, B A must appear together.

Let's assume A is the fifth value, in the case

_ _ C B A

In this case, for the second and first position we can have either C B or B A or C C.

Let's assume B is the fifth value, in the case

_ _ _ C B

The code can be

C B A C B

A C B C B

B C B C B

C B A C B

In all cases we have more than one C. Hence B cannot be the fifth value.

Let's assume C is the fifth value, in the case

_ _ _ _ C

If B were appear in any one of the 4 places, and C must precede B or the code can be C C C C C. However in both cases we have more than one C in the code.

Hence, the only possibility is A is the fifth digit.

The statement alone is sufficient.

Option B
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 10:11
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Bunuel
12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A certain five-letter code may only be constructed with A’s, B’s, and C’s. If an A appears in any position but the first, it must be immediately preceded by a B. If a B appears in any position but the first, it must be immediately preceded by a C. What is the fifth letter in the code?

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
(2) There is exactly one C in the five-letter code.


 


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for the 12 Days of Christmas Competition

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__ __ __ __ __ be the 5 boxes
L1 L2 L3 L4 L5

Given that :
  • Rule 1 : BA should be the order (except when A is in L1, makes sense)
  • Rule 2: CB should be the order (except when B is in L1, makes sense)
  • L5 = ?

S1 :

  • L1 = L4
  • L2 =L5

take L1 = C,
seq becomes C __ __ C __
then, seq becomes C B A C B (complying with rules Rule 1 and Rule 2)
or C C C C C for that matter

insufficient

S2:

only one C
C B B B B
C B A A A

Insufficient

Both :
  • we cant take L1 = C or L2 = C since only one C is present
  • say L1 = A
  • A __ __ A __
  • A C B A C is only option (since L2 = L5) but this violates exactly one C condition
  • say L1 = B
  • B __ __ B __
  • B A C B A
  • This satisfies all condition
  • L2 = A
    • __ A __ __ A
    • B A C B A
    • (same as above case)
  • L2 =B
    • __ B __ __ B
    • C B __ C B
    • violates exactly one C condition
Thefore C is answer IMO
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 10:14
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Five letter code: _ _ _ _ _
Let the positions be denoted by 1,2,3,4,5
We know:
If an A appears in any position but the first, it must be immediately preceded by a B
For ex: if A appears in 2nd position, B must be in first position
If a B appears in any position but the first, it must be immediately preceded by a C.
For ex: if B appears in 2nd position, C must be in the first position

We need to find the 5th position.

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
If position 1 is A, position 4 is also A. Here, position 2 can be anything, B or C. We don't know. Insufficient

(2) There is exactly one C in the five-letter code.
C can be anywhere in the 5th position or elsewhere, hence insufficient

Combining both (1) and (2);
Only 1 C, and we know position 1= position 4 and position 2= position 5 => C can only be in the 3rd position.
If C is in the 3rd position, we know only possible combination is BACBA hence fifth letter is A.
Both statements together are sufficient (C)
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 10:23
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Given the constraints,

Statement 1: Possible combinations: ACBAC or BACBA. 2 possible values, Insufficient
Statement 2: Possible combinations: BACBA, there's no other value possible, because the minute we place A or B anywhere but the first, the constraints trigger. Hence, answer is B
Bunuel
12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A certain five-letter code may only be constructed with A’s, B’s, and C’s. If an A appears in any position but the first, it must be immediately preceded by a B. If a B appears in any position but the first, it must be immediately preceded by a C. What is the fifth letter in the code?

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
(2) There is exactly one C in the five-letter code.


 


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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 10:34
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Bunuel
12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A certain five-letter code may only be constructed with A’s, B’s, and C’s. If an A appears in any position but the first, it must be immediately preceded by a B. If a B appears in any position but the first, it must be immediately preceded by a C. What is the fifth letter in the code?

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
(2) There is exactly one C in the five-letter code.


 


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for the 12 Days of Christmas Competition

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Fifth letter?

Statement 1 - The first and fourth letters in the code are the same, as are the second and fifth letters.
Possible codes - BACBA, ACBAC, BCCBC
Last letter can be A or C, so not sufficient.

Statement 2 - There is exactly one C in the five-letter code.
Possible code - BACBA
Last letter is A, so it's sufficient

Answer: B
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 10:38
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For A: At any position other than first B is immediately ahead of A
For B: At any position other than first C is immediately ahead of B
For C: No rules

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
According to this we get below combinations:
- CBACB
- BACBA
- ACBAC

So, in these 3 combinations only there is different letter at 5th position
Not sufficient.

(2) There is exactly one C in the five-letter code.
According to this we get only 1 combination: BACBA
For letters B and C at 5th position combination will not satisfy the condition for C to be one time only.

Sufficient.

Answer: B
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 10:45
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Quote:
12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A certain five-letter code may only be constructed with A’s, B’s, and C’s. If an A appears in any position but the first, it must be immediately preceded by a B. If a B appears in any position but the first, it must be immediately preceded by a C. What is the fifth letter in the code?

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
(2) There is exactly one C in the five-letter code.


 


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for the 12 Days of Christmas Competition

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Rules:
A must be preceded by B (BA).
B must be preceded by C (CB).
Goal: Determine the fifth letter.
(1) First and fourth letters are the same; second and fifth letters are the same (X Y _ X Y).
If X=C: We have C Y _ C Y. Y must be B (CB). Then, the third letter must be A (CBA). So, the code is CBACB. The fifth letter is B.
If X=B: We have B Y _ B Y. Y must be A (BA). Then, we have BA _ BA. But B must be preceded by C. Impossible.
If X=A: We have A Y _ A Y. Y must be preceded by B, which must be preceded by C. Impossible.
Therefore, only CBACB works. The fifth letter is B. Sufficient.
(2) Exactly one C.
If there's only one C, the code could be CBAAA (fifth letter A) or CBBBB (fifth letter B). Insufficient.
Conclusion: Only (1) is sufficient.

IMO A
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 10:46
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The stem tells us that;
  • A certain five-letter code may only be constructed with A’s, B’s, and C’s
  • If an A appears in any position but the first, it must be immediately preceded by a B
  • If a B appears in any position but the first, it must be immediately preceded by a C

Then it asks us;
  • What is the fifth letter in the code?

Let's understand the information first;

_ _ _ _ _ These spaces will be filled by a,b or c.

A _ _ _ _ B A _ _ _
B A _ _ _ C B A _ _

Let's look at the statements;

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

If this is the case, then

A C B A C
C B A C B
B A C B A

Statement 1 is not sufficient

(Eliminate A and D)

Let's look at statement 2;

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

Only possible case is B A C B A

Statement 2 is sufficient.

(Eliminate CE)

Answer is B
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 10:55
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A certain five-letter code may only be constructed with A’s, B’s, and C’s. If an A appears in any position but the first, it must be immediately preceded by a B. If a B appears in any position but the first, it must be immediately preceded by a C. What is the fifth letter in the code?

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
(2) There is exactly one C in the five-letter code.




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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 10:59
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The answer is that together it's sufficient. C

Statement 1: we can have any combination. so insufficient.

Statement 2: We have only one C and it can have any combination, so insufficient.

Combining both the statements, we have only one combination of code which is ABCAB.
Thus together sufficient.
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 11:15
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B
Rules:
  1. The code consists of only A, B, and C.
  2. If an A appears in any position other than the first, it must be immediately preceded by a B.
  3. If a B appears in any position other than the first, it must be immediately preceded by a C.
(1)
when first and fourth letters are the same the possible combinations can be
BACBA
ACBAC
CBACB
insuffiecient as we cant certainely determine the fifth letter

(2)
There is exactly one C in the five letter code
The only combination can be where C is in the middle which gives us only one combination which is
BACBA

here we can certainly say that A is the fifth letter
Hence, this statement alone is sufficient
Bunuel
12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A certain five-letter code may only be constructed with A’s, B’s, and C’s. If an A appears in any position but the first, it must be immediately preceded by a B. If a B appears in any position but the first, it must be immediately preceded by a C. What is the fifth letter in the code?

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
(2) There is exactly one C in the five-letter code.


 


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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 11:22
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A certain five-letter code may only be constructed with A’s, B’s, and C’s. If an A appears in any position but the first, it must be immediately preceded by a B. If a B appears in any position but the first, it must be immediately preceded by a C. What is the fifth letter in the code?

Solution:
(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
So there's only 1 letter in the middle. We don't know the numbers. Insufficient.

(2) There is exactly one C in the five-letter code.
If numbers are consequent, and C is in the middle. That means B is before C and in the fifth place. Sufficient.
B is correct.
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12 Days of Christmas GMAT Competition - Day 8: A certain five-letter 25 Dec 2024, 11:59
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The easier option is to write down the possible outcomes
From statement 1 we can have code BACBA or CBACB so not sufficient
From statement 2 knowing only one C is available is insufficient
Combining the two statements we can conclude that the code is BACBA
Choice C
Bunuel
12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A certain five-letter code may only be constructed with A’s, B’s, and C’s. If an A appears in any position but the first, it must be immediately preceded by a B. If a B appears in any position but the first, it must be immediately preceded by a C. What is the fifth letter in the code?

(1) The first and fourth letters in the code are the same, as are the second and fifth letters.
(2) There is exactly one C in the five-letter code.


 


This question was provided by Manhattan Prep
for the 12 Days of Christmas Competition

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