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

1 2 3

Nsp10

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26 Dec 2024, 04:05

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statement 1.
we can have three cases
1. Starting with A {and since the first letter is A we also know 4th letter which is also to be A} A _ _ A _ {we have to place B before 4th A because given} A _ B A _
and we have to mark C before B then the code would be ACBAC
2. Starting with B
3. Starting with C

Case 2 and Case 3 would produce different code so Statement 1 is not sufficient.

Satement 2.

with exactly one C we can produce many such codes
eg ABACB or CBABA etc

on combining we get 2 conditions now
1st is The first and fourth letters in the code are the same, as are the second and fifth letters. and There is exactly one C in the five-letter code.

so possible code could be BACBA.

Bunuel

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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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Prakhar9802

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Five letter code = _ _ _ _ _

If A appears in any position but first, B should come just before A.
If B appears in any position but first, C should come just before B.

We want to know the last letter (fifth).

STATEMENT 1 -

1st and 4th letters are same. So are the 2nd and 5th letter.

The code can be -

CCCCC = last letter C
or
CBACB = last letter B

Therefore, INSUFFICIENT.

STATEMENT 2 -

exactly 1 C in the code.

The only possible combination with 1 C and the given restrains is -

BACBA = last letter A

SUFFICIENT.

Therefore, FINAL ANSWER - Option B

Bunuel

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Bunuel

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Statement (1):
The first and fourth letters in the code are the same, as are the second and fifth letters.
Let the code be X1 X2 X3 X4 X5. From this statement:

X1=X4,
X2=X5.

This imposes some symmetry on the code, but we cannot deduce X1, X2, or X3 uniquely without further information. For example:

X1=A, X2=C, and X3=B are valid under the rules, leading to ACBAC, with X5=C.
X1=B, X2=A, and X3=C could lead to BACBA, with X5=A

Without further constraints, X5 cannot be determined uniquely.
Statement (1) alone is insufficient.

Statement (2):
[b]There is exactly one C in the five-letter code.[/b]
If there is only one C, it must be positioned in a way that allows the rules to be satisfied:

A B must always follow a C if B is not in the first position.
An A must always follow a B if A is not in the first position.

Let's try various positions for C
If we start with C, there is not possible combination, (CBA, the next two values post A can not be A or B according to the constraints, and we cannot add C due to the statement (2) condition)
If the second letter is C, there is not possible combination, (BCBA or ACBA, the next value post A can not be A or B according to the constraints, and we cannot add C due to the statement (2) condition)
If the third letter is C, there is not possible combination, (BACBA), therefore the 5th letter has to be A
Similarly, If the fourth letter or fifth letter is C, there is not possible combination

Therefore, only Statement (2) is sufficient.

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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.

Multiple possibilities exist

e.g bacba ( a is 5th letter in the Code)
acbac ( c is 5th letter in the Code)

answer can't be uniquely determined

INSUFFICIENT

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

[color=#000000] e.g. bacba ( a is 5th letter in the Code) [/color]

[color=#000000]Only one option is possible[/color]

[color=#000000]answer can be uniquely determined [/color]

SUFFICIENT

(B) is the CORRECT answer

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26 Dec 2024, 04:49

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Start this question with the rules given

Consider Statement A
1st and 4th digit is same and 2nd and 5th Digits are same

start with A

A _ _ _ A _
ACBAC (C is on the 5th place)

Start With B
BACBA (A is on the 5th)

So A is Insufficient

Consider Only Statement B

Start with A
A C B A (we can't put C as it's only one time available), so Rejected Case

Start With B
B A C B A (getting the pattern), so yeah possible
B C B A (no further Possible)

Start with C
C B A (then No further Possible)

So B is sufficient

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26 Dec 2024, 05:06

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The correct answer is C

Question: What is the fifth letter in the code?

Given: A is immediately preceded by B i.e the sequence will be BA
B is immediately preceded by C i.e the sequence will be CB

Combining both, we get the sequence CBA

We know that A and B cannot come in the first position, so possible combinations are:

CBA_ _

_ CBA _

_ _ CBA

Statement 1:
It is given that the first and fourth letters in the code are the same, and the second and fifth letters are the same

CBA C B

A CBA C

B A CBA

From above, we see that the fifth letter can be either A, B, or C - Insufficient

Statement 2:
Given: There is exactly one C in the five-letter code - Insufficient as it does not give us any details about the letter in the fifth position

Combining 1 and 2: Only the third sequence is possible. The fifth letter will be A - Sufficient

CBA C B

A CBA C

B A CBA

Bunuel

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26 Dec 2024, 06:07

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S1 gives multiple cases.

S2 states only one C
we have

BACBA

We can get the fifth letter

B)

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Bunuel

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As A and B appears in any position but first, C must be the first letter
B must be after C and A must be after B, so BA must be somewhere after C.

(1) As first and fourth letter are the same so

C _ _ C _

So BA must be on 2nd and 3rd letter

C B A C _

As second and fifth letter are the same

C B A C B

B will be the fifth letter in the code (Sufficient)

(2) As there is exactly one C in the five-letter code and B must be after C and A must be after B

C B A _ _

As the condition of B after C is not valid, so the condition A after B is not valid for the remaining

So last two digit can be either A or B (Insufficient)

Therefore the answer is A

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Statement1-

let the code be abcde-
so, a=d and b=e

therefor, abcab

Now, c can be A/B/C, keeping in mind the given rules.
knowing a and b allows us to determine e, which equals b.

However, we still cannot uniquely determine e.

So, statement1 is insufficient.

Statement2-
exactly 1 C in code.

So, if C appears it must precede B.

C can be at positions- a,c

This also doesn't uniquely determine the 5th letter.

So, statement 2 is also insufficient.

Combining the 2 statements-

Case1- a=C
therefor, CCcCC.
the above is invalid as C cannot precede another C.

Case2- b=C
therefor, aCcaC
Valid only if c adheres to the rule. If c=B, the preceding letter must be C.

Case3- c=C
therefor, abCab
if c=C, remaining letters a and b must be A and B respectively.
Combining these observations e=b, which is uniquely determined.
therefor, 5th letter is C.

Ans- C

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available choices A,B,C
- should be BA (only exception if code starts with A)
- should be CB (only exception if code starts with B)

(I)

BA C BA possible
CB A CB possible
AC B AC possible

fifth letter is not constant
Insufficient

(II)

only 1 C

if we have C, it will be followed by B, which will be followed by A
Let put that step by step
C B A B A possible
_ C B A _ not possible
B A C B A possible
_ _ _ C B not possible

A is the last letter
Sufficient

Option B

Bunuel

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rns2812

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5 letter code with A, B, C is ()()()()()

Given

If First - B/C, A is to be preceded by B
If First is A/C, B is to be preceded by C

Ask : What is the fifth letter in the code?

Statement 1 : 1st and 4th are same, 2nd and 5th are same

Possibilities when B/C is the first

(B/C) ( ) ( ) (B/C) ( ) -

(B) (A ) (B/C) (B) (A) or (C) (B/C) (A/B/C) (C) ( B/C)

Not sufficient

Statement 2 : Only 1 C is there

(B)(A/B/C)(B/C/A)(B/C/A)(B/C/A) or (C)(B)(A)(Not possible)(A) or (A)(A/C)(A/C/B)(A/C/B)(A/C/B)

Not sufficient as the last digit will depend on the position of C

Together : C will be 3rd

Only possible cases :
(B) (A) (C) (B) (A) or (A) (A) (C) (A) (A)

Sufficient - OPTION C

Bunuel

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26 Dec 2024, 08:04

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It is given that,
If A appears in any position except the first, B must immediately precede it
If B appears in any position except the first, C must immediately precede it

Statement 1
First and fourth letters are the same
Second and fifth letters are the same

There are multiple ways in which the letters can be arranged with the given information

Statement 1 alone is not sufficient

Statement 2

There is 1 C

There is only one way in which the letters can be arranged with the given conditions:

B A C B A

If C is put in any other position, the two given conditions will not hold true, for example, if C is the first letter,
C B A B A

The B in fourth place is nor preceded by C

Therefore, Statement 2 alone is sufficient

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26 Dec 2024, 08:11

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This is a little trick Question, lets start with possible cases

ACBAC
BACBA
BACCC
etc

(1) 1st and 4th are same 2nd and 5th are same
ACBAC
BACBA

5th digit C or A
Not sufficient

(2) There is exactly 1 C

Lets try > C as 1st place > C_ _ _ _

Since C is 1st> B precede A and C precede B so CBA(3rd digit can only be A) (4th digit cannot be C, not B since B must be preceded by C, not A since A must be preceded by b)

C as 2nd place > _ C _ _ _ > ACBAX Not possible, BCBAX Not Possible

C as 3rd Place > _ _ C _ _ > If a is 1st 2nd can't be A or B so 1st place must be B, BACBA Only possible

C as 4th place > _ _ _ C _ > Not possible

C as 5th Place > _ _ _ _ C> Not Possible

So it gives only BACBA

Sufficient to ans fifth character

Ans B

Bunuel

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The code can only consist of the letters A, B, and C.
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.

1->
1st letter must be A, B, or C
4th letter is determined once 1st letter is known
5th letter = 2nd letter

So the possible variations are
ACBAC, BACBA and CBACB

2-> There is exactly one C in the five-letter code
Multiple options are possible among these, so we cannot use this statement alone to determine the 5th letter specifically

Now using statement 1 and 2,
We are able to narrow down to 1 option out of the 3 in statement 1 and i.e,
BACBA

Which gives us the fifth letter to be A.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

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Bunuel

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5-letter code - only using A's, B's and C's
A: 1st position
any other position then BA
B: 1st position
any other position then CB
C: any position, no restrictions

Statement 1: 1st and 4th letters are the same, 2nd and 5th letters are the same.

The possible combinations are the following:
ACBAC
BACBA
CBACB

Since, there are multiple answers possible for the 5th letter in the code Statement 1 is not sufficient.

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

Possible combination
BACBA

Only one combination is possible if there exactly one C in the 5 letter word.

Statement 2 is sufficient.

[b] is the correct answer.

[/b]

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Bunuel

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(i)

the possible combinations according to the conditions are
ACBAC
BACBA
CBACB

We are getting 3 differnt values for fifth digit
(I) is not sufficient

(II)
the possible combinations according to the conditions are
BACBA
CBABA
if we try other combinations, they do not satisfy the conditions
So (II) is sufficient

Answer is Option B

Bunuel

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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?

As per given conditions, Sequence must be in order of -C-B-A-

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

Possible combinations
_C_ _B_ _A_ _C_ _B_

_B_ _A_ _C_ _B_ _A_

_A_ _C_ _B_ _A_ _C_

Different values of 5th letter. Insufficient.

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

Possible combinations
_B_ _A_ _C_ _B_ _A_

Unique value of 5th letter. Sufficient.

Ans. B