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Intern  Joined: 20 Sep 2011
Posts: 23
Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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I used the slot method to answer this one.
XXXX - First slot you have three options, second slot you have three options (nothing says you can't repeat letters), third slot you have two options and fourth slot you have two options to make sure that you include at least all of the letters. 3*3*2*2 = 36
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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

So, the 4-letter code will have A,B,C and a repeat letter from either A,B or C.
Our possible selections could be: {A,A,B,C}, {B,B,A,C}, and {C,C,A,B}

$$A,A,B,C --> 4!/2! = 12$$
$$B,B,A,C --> 4!/2! = 12$$
$$C,C,B,A --> 4!/2! = 12$$

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Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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First pick which letter is doubled. There are 3 ways. Without loss of generality, A is doubled.
Then pick a place for B. There are 4 ways to pick a place for B.
Then pick a place for C. There are 3 ways to pick a place for C.
Then place the two A's.

3*4*3=36.

It is similar to AKProdigy87's solution, except that we don't even need the 4C2. No formula whatsoever, just multiplication.
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Re: PS - Prob. of A 4-letter code  [#permalink]

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AKProdigy87 wrote:
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

What I don't get about this approach is why we don't multiply by 2! to account for the different permutations of YZ, and hence have an answer of 72.

I know it's an old problem, but would someone care to explain?

Where is this permutation of 2! for YZ already accounted for in this problem? That's really unclear to me
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Re: PS - Prob. of A 4-letter code  [#permalink]

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manimgoindowndown wrote:
I know it's an old problem, but would someone care to explain?

Where is this permutation of 2! for YZ already accounted for in this problem? That's really unclear to me

AABC, ABAC, ABCA, BAAC, BACA, BCAA, AACB, ACAB, ACBA, CAAB, CABA, CBAA
BBAC, BABC, BACB, ABBC, ABCB, ACBB, BBCA, BCBA, BCAB, CBBA, CBAB, CABB
CCAB, CACB, CABC, ACCB, ACBC, ABCC, CCBA, CBCA, CBAC, BCCA, BCAC, BACC

We agree that Y always preceeds Z. Then we have two ways to choose Y. For example, if B repeats twice, then we may have Y=A (e.g. BBAC) or Y=C (e.g. BBCA). Y and Z cannot be permuted.
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Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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Bunuel Please Clarify my doubt.

If the code includes all the three letters, then the 4th letter can any letter from ABCDEFG.... XYZ.

The question doesn't specify that the 4 letter code includes only A, B and C.

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
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Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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Rajkiranmareedu wrote:
Bunuel Please Clarify my doubt.

If the code includes all the three letters, then the 4th letter can any letter from ABCDEFG.... XYZ.

The question doesn't specify that the 4 letter code includes only A, B and C.

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

I think that it is specified.

We are told that a 4-letter code consists of letters A, B, and C and that the code includes ALL the three letters A, B, and C (so the case of AAAA is not possible).
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Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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My approach is the following:
aabc - 3!2! (aa as one unit, which gives 3!, b and c interchangeable, which is 2!)
bbac - 3!2!
ccab - 3!2!

12+12+12=36
Is it correct?
Thank you
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Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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Christy111 wrote:
My approach is the following:
aabc - 3!2! (aa as one unit, which gives 3!, b and c interchangeable, which is 2!)
bbac - 3!2!
ccab - 3!2!

12+12+12=36
Is it correct?
Thank you

First of all: 3!/2!=3, thus you'd have 3 + 3 + 3 = 9, not 12 + 12 + 12 = 36.

The number of ways to arrange AABC is 4!/2!=12. There is no need to consider AA as one unit, because we do not need AA to be together in each arrangement of AABC.

For complete solution check here: a-4-letter-code-word-consists-of-letters-a-b-and-c-if-the-59065.html#p782466

Hope it helps.
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Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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Another solution:
P(3,3)C(3,1)P(2,1)=36

(C)
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Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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Bunuel wrote:
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 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

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 $$3*\frac{4!}{2!}=36$$.

Hope it helps.

Could you please prove that "Total number of permutations for a set of 'n' objects of which 'r' objects are identical is n!/r!" or show me the link that explain the formulation n!/r!?
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Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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Bunuel wrote:
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 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

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 $$3*\frac{4!}{2!}=36$$.

Hope it helps.

Could you please prove that "Total number of permutations for a set of 'n' objects of which 'r' objects are identical is n!/r!" or show me the link that explain the formulation n!/r!?

Permutations of $$n$$ things of which $$P_1$$ are alike of one kind, $$P_2$$ are alike of second kind, $$P_3$$ are alike of third kind ... $$P_r$$ are alike of $$r_{th}$$ kind such that: $$P_1+P_2+P_3+..+P_r=n$$ is:

$$\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}$$.

For example number of permutation of the letters of the word "gmatclub" is 8! as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is $$\frac{6!}{2!2!}$$, as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be $$\frac{9!}{4!3!2!}$$.

Theory on Combinations: math-combinatorics-87345.html

DS questions on Combinations: search.php?search_id=tag&tag_id=31
PS questions on Combinations: search.php?search_id=tag&tag_id=52

Tough and tricky questions on Combinations: hardest-area-questions-probability-and-combinations-101361.html

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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

Just adding to everyone, especially for folks who use MGMAT books.
Please do correct me if I am wrong, I am still a noob.

Lets say we have to find number of ways the word MGMAT has to be arranged
For such a question, we use $$5!/2!$$

Rephrasing this question, lets say we are told there is a 5 Letter code made from the letters "M,G,A,T" in which one of the letters repeats once - we do not know which one.

Then we use the following calculation

Arrangement in case M is repeated OR Arrangement in case G is repeated OR Arrangement in case A is repeated OR Arrangement in case T is repeated

$$5!/2!$$ + $$5!/2!$$ +$$5!/2!$$ + $$5!/2!$$

Similarly, in this question we have 3 Letters in which we do not know which we know atleast one of them repeats but we do not know which one
So
Arrangement in case A is repeated OR Arrangement in case B is repeated OR Arrangement in case C is repeated

$$4!/2!$$ + $$4!/2!$$ +$$4!/2!$$ =36
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Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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Bunuel wrote:
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 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

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 $$3*\frac{4!}{2!}=36$$.

Hope it helps.

Can someone please explain why 4!/2! represent the number of ways to arrange 4 letters of which one is repeated twice?? Thank you!
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Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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happyface101 wrote:
Bunuel wrote:
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 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

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 $$3*\frac{4!}{2!}=36$$.

Hope it helps.

Can someone please explain why 4!/2! represent the number of ways to arrange 4 letters of which one is repeated twice?? Thank you!

THEORY:

Permutations of $$n$$ things of which $$P_1$$ are alike of one kind, $$P_2$$ are alike of second kind, $$P_3$$ are alike of third kind ... $$P_r$$ are alike of $$r_{th}$$ kind such that: $$P_1+P_2+P_3+..+P_r=n$$ is:

$$\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}$$.

For example number of permutation of the letters of the word "gmatclub" is 8! as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is $$\frac{6!}{2!2!}$$, as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be $$\frac{9!}{4!3!2!}$$.

For more check the links below:

Theory on Combinations

DS questions on Combinations
PS questions on Combinations

Tough and tricky questions on Combinations
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Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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can somebody solve it with a slot method ?
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Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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

Hi,

Four letter code, let's assume ABCX, X any letter "A or B or C".

As you want to solve this with slot method,

"ABC" can be arranged in 3!.

Now come to ABCX :-

A can be arranged in any 4 way.
Similarly B can arrange in any 3 way, C can arrange in 2 way and Last "X" which comprises of (A or B or C).
Let suppose if you put "A" again at forth place this can done in 1 way as one slot is remaining to fill
ABCA - > 4*3*2 *1 = 24.
But as "A" is repeated twice, we need to remove this ambiguity.
We divide this by 2, which is 24/2 = 12.

Now, Total combination is 12*3 = 36, which is correct answer.

Hope this will help you Thanks.
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A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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2
A 4 letter code without constraint i.e, with 4 unique letters can be formed in 4! ways.
Constraint is only 3 letters are unique and one of them repeats.
With constraint the answer is 4! /2! =12 ways
There are three such cases i.e., A ,B or C may repeat.
So the total number of ways = 12*3=36
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A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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First we need to think about the words that are going to repeat.

1st case: ABCA (you have two letters A)
4!/2! = 12 (2! is used because the letter A repeats 2 times)

2nd case: ABCB
4!/2! = 12

3rd case: ABCC
4!/2! = 12

12*3= 36
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Re: A 4-letter code word consists of letters A, B, and C. If the  [#permalink]

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1
Honestly, this is a really tough topic. I calculated it this way:

ABCX + ABXC + AXBC + XABC then divide by 2 because of the repeat (i.e. ABCC and ABCC would show up in the first term and the second term?

3*2*1*3 + 3*2*3*1 + 3*3*2*1 + 3*3*2*1 = 72 --> 72/2 = 36

Can you please tell me if this is a correct line of thinking. Or better yet, how to develop any type of intuition on these problems? Re: A 4-letter code word consists of letters A, B, and C. If the   [#permalink] 14 Dec 2015, 14:33

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