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Manager  B
Joined: 15 Dec 2015
Posts: 50
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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If the question didnt involve the restriction -"If the code includes all the three letters", would the answer be 3^4=81?
I am not able to imagine how that should be the answer. Can some one help? thanks
CEO  S
Joined: 20 Mar 2014
Posts: 2626
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44 GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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1
BrainLab wrote:
can somebody solve it with a slot method ?

If you use the slot method, you can clearly see that the possible combinations will be

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

Thus no matter what set you choose, you will invariably end up with 2 same letters in any given combination of 4 letters.

Thus permutations of XYZX = 4!/2! and as you have 3 options to select A or B or C for the repetitive letter, the total number of arrangements possible = 4!/2!*3 = 36.

D is thus the correct answer.

Hope this helps.
CEO  S
Joined: 20 Mar 2014
Posts: 2626
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44 GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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Nina1987 wrote:
If the question didnt involve the restriction -"If the code includes all the three letters", would the answer be 3^4=81?
I am not able to imagine how that should be the answer. Can some one help? thanks

3^4 is only possible if you are given that the 4 letter code can have repetitive letters such as AAAA or BBBB or CCCC etc.
Intern  Joined: 05 Apr 2016
Posts: 26
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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Can anyone help me with this one?
This is my reasoning:
Total combination of codes: 3*3*3*3=81
Cases of only two-letter codes: 2*2*2*2-2 (the -2 if for the cases that only 1 letter repeats 4 times)(could be any of the two letters).
Cases of one-letter codes: 3 (AAA, BBB or CCC).

Answer: 81-14-3= 64

Makes sense to me!
Intern  B
Joined: 27 Apr 2016
Posts: 1
Schools: Ryerson
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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A/c to the question,we have to make a 4 letter word using the letters A,B,C only and all letters have to be used.So,there are 3 possible cases:

Case 1: Using A twice,B once,C once

How many different codes can be formed using the letters A,A,B,C? = 4!/2!= 12

Explanation: N different objects can arrange in n different places in n! ways.So,these 4 letters arrange in 4 places in 4! ways if these all were different.Since, 2 letters are same ,the actual no. of arrangements would be 4!/2!

Case 2: Using B twice,A once,C once

Similar to Case 1,Total no. of arrangements would be 12

Case 3 : Using C twice,A once,B once

Similar to Case 1,Total no. of arrangements would be 12

So,Total arrangements would be 36

Hope that helps!
Director  S
Joined: 17 Dec 2012
Posts: 630
Location: India
A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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Top Contributor
GHIBI wrote:
A 4-letter code word consists of letters A, B, and C. If the code includes all the three letters, how many such codes are possible?

A. 72
B. 48
C. 36
D. 24
E. 18

Take one case: AABC

With two A's together, there are 6 ways of forming a code.

Separate A's by one letter. There are 4 ways of forming a code.

Separate A's by 2 letters . There are 2 ways of forming a code.

A total of 12 ways. Similarly for B and C repeating . Final total of 36.
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Manager  B
Joined: 02 Jul 2016
Posts: 108
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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Well the question is a bit ambiguous as it is not clear whether the 4 letter word is to formed ONLY by the 3 letters or the 4th letter can be different also
In the solution given below I have considered that the 4 letter word is formed only by the 3 letters.Hence in this case it obvious that each letter will be repeated twice for each combination
therefore
for
4!/(2!)*3=36 IS THE ANSWER
Manager  G
Joined: 09 Jan 2016
Posts: 104
GPA: 3.4
WE: General Management (Human Resources)
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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GHIBI wrote:
A 4-letter code word consists of letters A, B, and C. If the code includes all the three letters, how many such codes are possible?

A. 72
B. 48
C. 36
D. 24
E. 18

A bit logic is needed for this problem.there are three letters and we need to form 4 letters code.
_ _ _ (A,B,or C).For every code we have to repeat one letter.

So, the solution will be : 3*4/2! = 36 .
hope it helps
Target Test Prep Representative G
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2823
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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GHIBI wrote:
A 4-letter code word consists of letters A, B, and C. If the code includes all the three letters, how many such codes are possible?

A. 72
B. 48
C. 36
D. 24
E. 18

We are given three letters, A, B, and C, and we must create a four-letter code in which all three letters are used. So, one letter must be repeated. Thus, we have the following three options:

1) A-B-C-A (if A is repeated)

2) A-B-C-B (if B is repeated)

3) A-B-C-C (if C is repeated)

Let’s start with option 1:

We see that there are four total letters and two repeated As. Thus, that code can be selected in the following number of ways:

4!/2! = (4 x 3 x 2 x 1)/(2 x 1) = 4 x 3 = 12 ways

Since the second code, A-B-C-B, has two Bs rather than two As, we can create the second code in 12 ways. Likewise, since the third code, A-B-C-C, has two Cs rather than two As or two Bs, we can create the third code in 12 ways.

Thus, the code can be created in 12 + 12 + 12 = 36 ways.

Answer: C
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Intern  B
Joined: 26 Mar 2017
Posts: 2
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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Bunuel wrote:
pavanpuneet wrote:
Here is how I tried to solve the question:

Consider XXXX = Assume the first three position is taken as for letters ABC those can be filled in 3! ways and then last letter can be filled in 3 ways... thus a total 18 ways.

Next, assume, that it XABC = 18 ways; next. CXAB = 18 ways; next BCXA = 18 ways... thus a total of 18*4 = 72ways!

Note that the correct answer to this question is 36, not 72.

A-ABC can be arranged in 4!/2!=12 ways;
B-ABC can be arranged in 4!/2!=12 ways;
C-ABC can be arranged in 4!/2!=12 ways;

Total: 12+12+12=36.

Answer: C.

Sorry but I am really struggling to understand this:

-ABC can be arranged in 4!/2!=12

Many thanks
Math Expert V
Joined: 02 Sep 2009
Posts: 55272
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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1
Pol DC wrote:
Bunuel wrote:
pavanpuneet wrote:
Here is how I tried to solve the question:

Consider XXXX = Assume the first three position is taken as for letters ABC those can be filled in 3! ways and then last letter can be filled in 3 ways... thus a total 18 ways.

Next, assume, that it XABC = 18 ways; next. CXAB = 18 ways; next BCXA = 18 ways... thus a total of 18*4 = 72ways!

Note that the correct answer to this question is 36, not 72.

A-ABC can be arranged in 4!/2!=12 ways;
B-ABC can be arranged in 4!/2!=12 ways;
C-ABC can be arranged in 4!/2!=12 ways;

Total: 12+12+12=36.

Answer: C.

Sorry but I am really struggling to understand this:

-ABC can be arranged in 4!/2!=12

Many thanks

There are 4 letters not 3. For example, it says A-ABC can be arranged in 4!/2!=12 ways. AABC, so 4-letter out of which two A's are identical can be arranged in 4!/2!=12 ways.
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Intern  Joined: 23 Dec 2017
Posts: 2
A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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If a 4 letter code has all the 3 letters A B and C, that means that 1 letter is repeated.
Let's say it is repeated in position 1 and 2, the number of possibilities is 3 * 1 * 2 * 1= 6, because once we have the first letter, we only have one possibility for the second one.
Now we only need to know how many combinations of 2 positions we can have in 4 positions = C(4,2) =6
So the answer is 6 * 6 = 36
C
Intern  B
Joined: 30 Nov 2015
Posts: 29
Location: United States (CO)
WE: Information Technology (Consumer Products)
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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It took me a little while to digest this one. Here is how i understood it-

I broke the question into 2 parts-
1. Select the letters
AND
2. Arrange the letters

1. Select the letters- we know that 3 of the 4 digits have to be A B C. They can be selected in 1 way (3C3). The 4th digit is going to be one of 3 (A B C) which can be selected in 3 ways (3C1). Hold onto this.

2. Arranging the letters- Now we have 4 digits that need to be arranged. We also know that one of these digits is repeating itself twice. It will be 4!/2! = 12.

Now we just use AND to combine these two. 1*3*12 = 36.
Intern  Joined: 30 Sep 2018
Posts: 6
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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Idea : 3 Letters but 4 places to be arranged. So there has to be a repetition of one of the letters exactly once assuming the code words does not contain any other elements.

Approach :
What is the output : Code word. so order matters.Also there are 4 positions. So we need to use combinations for each possibility of placement of three letters.
So there are 4C3 ways of placing these three letters among these 4 places and in each case order matters , so 3! ways of arranging these three letters . Now the letter in the fourth position in each case has to be one of three letters so we have three choices.So total possible cases are:
4C3 * 3! * 3 = 72 ways.
However we will have repetition of one letter in each cases so by symmetry property we have to slice it into 2! ways.
So the number of distinct possible code words: 72 / 2 = 36 ways.
Senior Manager  G
Status: Gathering chakra
Joined: 05 Feb 2018
Posts: 253
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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To visualize: The trap to avoid is that you don't only have to multiply the 3P3 by 3 possibilities for the 4th letter (which would be 18) but also multiply by 2 for the position (either in front or behind).
Intern  B
Joined: 19 Jan 2018
Posts: 49
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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GHIBI wrote:
A 4-letter code word consists of letters A, B, and C. If the code includes all the three letters, how many such codes are possible?

A. 72
B. 48
C. 36
D. 24
E. 18

Well we know that we have to make a 4 letter code word with only 3 letters (A, B and C). That means we'll have a duplicate letter

I like to visualize in this manner _ ,_ ,_ , _
Because there's a duplicate letter, we can arrange it in 3*3*2*2 as order does matter. So the answer is 36

C is the answer
Intern  B
Joined: 22 Sep 2018
Posts: 9
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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GHIBI wrote:
A 4-letter code word consists of letters A, B, and C. If the code includes all the three letters, how many such codes are possible?

A. 72
B. 48
C. 36
D. 24
E. 18

It's easy to get tricked into thinking the sequence of the four letter code as

ABCX

In this situation ABC can be arranged in 3! ways and X in 3 ways. So the no of ways is 3! * 3 =18 ways..but

AABC or ABBC is possible too.so the above way of solving the problem is wrong.

no of ways of arranging 3 letters in four slots is 4!/2! ( 2 as one letter will repeated in the sequence like AABC) [this is for one number]

and there are 3 letter that can be repeated in the sequence so... 3* 4!/2! = 36

Hope I am clear!!

Cheers... CEO  P
Joined: 18 Aug 2017
Posts: 3524
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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GHIBI wrote:
A 4-letter code word consists of letters A, B, and C. If the code includes all the three letters, how many such codes are possible?

A. 72
B. 48
C. 36
D. 24
E. 18

A,B,C are letters and code has to have all three letters
3*3*2*2 ; 36
IMO C
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If you liked my solution then please give Kudos. Kudos encourage active discussions. Re: A 4-letter code word consists of letters A, B, and C. If the   [#permalink] 14 Apr 2019, 10:52

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