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A 4letter code word consists of letters A, B, and C. If the
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25 Jan 2008, 23:19
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A 4letter 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
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Re: PS  Prob. of A 4letter code
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15 Sep 2010, 15:52
MBAwannabe10 wrote: here is my approach: 3*2*1*3 = 18 but there is 4!/2! of arranging them =>36 ways in the end I don't see how the above way is giving 36 as an answer. A 4letter 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 As code must include all the three letters then pattern of the code word is ABCX where X can be any letter out of A, B, and C. So we can have the code word consisting of letters: ABCA; ABCB; ABCC. We can arrange letters in each of above 3 cases in \(\frac{4!}{2!}\) # of ways (as each case has 4 letters out of which one is repeated twice), so total # of code words is \(3*\frac{4!}{2!}=36\). Answer: C. Hope it helps.
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Re: PS  Prob. of A 4letter code
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26 Jan 2008, 07:23
It has to be 36 .. We know that the code already contains the the letters A,B and C. Now the 4th letter can be choosen in 3 ways (A,B,C). Once we have the set of 4 letters we can arrange them in 4!/2! ways. [Total number of permutations for a set of 'n' objects of which 'r' objects are identical is n!/r!. if there are multiple groups of identical objects like say r1 objects which are red and r2 objects which are green .. then the total permutations would be n!/r1!r2! ...] Now the total ways of arranging would be 4!/2!*3 = 4*3*3 = 36 walker wrote: if you think N=36, try to add a new 3letters code to the 18set:
ABCA,ABCB,ABCC ACBA,ACBB,ACBC BACA,BACB,BACC BCAA,BCAB,BCAC CABA,CABB,CABC CBAA,CBAB,CBAC How about AACB ? ANSWER:C




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Re: PS  Prob. of A 4letter code
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26 Jan 2008, 00:11
E\(N=C^3_1*C^2_1*C^1_1*C^3_1=3*2*1*3=18\) or \(N=3^4C^3_1C^3_1*C^2_1*C^4_1C^3_1*C^4_2*C^2_1*C^2_2=813*2*43*6*2*1=8132436=18\) or \(N=P^3_3*C^3_1=3*2*3=18\)
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Re: PS  Prob. of A 4letter code
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26 Jan 2008, 01:59
E: 18
3 letters A, B and C can be arranged in 3! ways = 6 ways 4th letter can be chosen in 3 ways
Total number of ways = 6*3=18



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Re: PS  Prob. of A 4letter code
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26 Jan 2008, 02:20
srp wrote: E: 18
3 letters A, B and C can be arranged in 3! ways = 6 ways 4th letter can be chosen in 3 ways
Total number of ways = 6*3=18 but then you assume that 4th letter can stand only in the end, don't you?



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Re: PS  Prob. of A 4letter code
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26 Jan 2008, 02:30
no, its not A. will be back in few moments
THE ANSWER SHOULD BE 36 WHICH IS HALF OF 72. CODE IS ESSENTIALLY A PERMUTATION, A1A2BC SHOULD BE THE SAME AS A2A1BC
3!*3*4/2= 36



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Re: PS  Prob. of A 4letter code
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26 Jan 2008, 02:45
CaspAreaGuy wrote: srp wrote: E: 18
3 letters A, B and C can be arranged in 3! ways = 6 ways 4th letter can be chosen in 3 ways
Total number of ways = 6*3=18 but then you assume that 4th letter can stand only in the end, don't you? Right! so we then get ABC[R] where R stands for repeat letter of type A,B,C Number of ways of arranging ABC[R] = 4!/2! and since we have to do this 3 times for each A,B and C, Total ways = 4!/2! * 3 = 36



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Re: PS  Prob. of A 4letter code
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26 Jan 2008, 04:52
if you think N=36, try to add a new 3letters code to the 18set: ABCA,ABCB,ABCC ACBA,ACBB,ACBC BACA,BACB,BACC BCAA,BCAB,BCAC CABA,CABB,CABC CBAA,CBAB,CBAC
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Re: PS  Prob. of A 4letter code
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26 Jan 2008, 05:07
Hi, CaspAreaGuy, nice to see you here CaspAreaGuy wrote: srp wrote: E: 18
3 letters A, B and C can be arranged in 3! ways = 6 ways 4th letter can be chosen in 3 ways
Total number of ways = 6*3=18 but then you assume that 4th letter can stand only in the end, don't you? I think that position of 4th letter doesn't matter 3* 2 * 1* 3=18 3* 3 * 1* 2 =18 3* 3* 2 * 1 =18



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Re: PS  Prob. of A 4letter code
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26 Jan 2008, 09:13
You are right! +1 for Q and +1 for AACB live and learn....
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Re: PS  Prob. of A 4letter code
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26 Jan 2008, 10:15
I corrected my formula: C\(N=\frac{C^3_3*C^3_1*P^4_4}{P^2_2}=\frac{1*3*4*3*2}{2}=36\) or \(N=3^4C^3_1C^3_1*C^2_1*C^4_1\frac12 *C^3_1*C^2_1*C^4_2=813*2*4\frac12 *3*6*2*1=8132418=36\)
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Re: PS  Prob. of A 4letter code
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26 Aug 2008, 07:56
GHIBI wrote: A 4letter 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 ABCA + ABCB + ABCC = 4!/2! +4!/2!+4!/2! = 36
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Re: PS  Prob. of A 4letter code
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05 Sep 2009, 13:05
36 for me too. The group of 4 letters would be ABCX where X is A/B/C just find out the permutation for 1 specific case say ABCA 4 things can be permuted in 4! ways and since 2 things are same(here 2 A's) divide by 2! therefore... 4!/2! Since there are three such groups based upon value of X , multiply by 3. Ans: 3*(4!/2!) = 36
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Re: PS  Prob. of A 4letter code
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10 Sep 2009, 20:48
(4!/2) + (4!/2) + (4!/2 ) = 3*(4!/2 ) = 36 .
Ans: C
Explanation:
Assuming A is the repeated letter, we get the 1st 4!/2, OR if B is the repeated letter, we get the 2nd 4!/2 OR if C is the repeated letter, we get the 3rd 4!/2
that gives (4!/2) * 3 = 36



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Re: PS  Prob. of A 4letter code
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11 Sep 2009, 15:34
I get C: 36 as well. This is how I approached the problem:
The 4 letters can be distinguished as follows: X  the letter which is duplicated. Y and Z  the two remaining letters, with Y always preceding Z in the code word.
As a result, a code word looks like XXYZ, or XYXZ, etc.
The number of possible combinations is as follows:
4C2  choose 2 of the 4 character places to put the duplicate characters (X in this case) * 3!  3 ways to choose X, 2 ways to choose Y, 1 way to choose Z.
4C2 * 3! = 36



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Re: PS  Prob. of A 4letter code
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15 Sep 2010, 15:15
here is my approach: 3*2*1*3 = 18 but there is 4!/2! of arranging them =>36 ways in the end
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Re: A 4letter code word consists of letters A, B, and C. If the
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21 Jun 2012, 01:51
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!



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Re: A 4letter code word consists of letters A, B, and C. If the
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21 Jun 2012, 01:58



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Re: A 4letter code word consists of letters A, B, and C. If the
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21 Jun 2012, 23:39
arvind410 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. AABC can be arranged in 4!/2!=12 ways; BABC can be arranged in 4!/2!=12 ways; CABC can be arranged in 4!/2!=12 ways; Total: 12+12+12=36. Answer: C. A quick question. In the above case , we assume that the extra letter is always in the first position , i.e AABC, BABC. Why aren't we accounting for ABCA , ABCB . And since it is a code, shouldn't arrangement matter, as in ABC is different from BAC ? Do correct me if I am wrong . We are not assuming that at all. The solution above says: AABC can be arranged in 4!/2!=12 ways. So, for the case when we have 2 A's in a code, 12 different arrangements are possible: AABC; ABAC; BAAC; ... The same for other cases. Hope it's clear.
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Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
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