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# How many combinations of three letters taken from letters

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SVP
Joined: 21 Jan 2007
Posts: 1856
Location: New York City
How many combinations of three letters taken from letters  [#permalink]

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02 Dec 2007, 13:29
5
40
00:00

Difficulty:

95% (hard)

Question Stats:

29% (02:18) correct 71% (02:07) wrong based on 551 sessions

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How many combinations of three letters taken from letters (a, a, b, b, c, c, d) are possible?

A. 12
B. 13
C. 35
D. 36
E. 56
##### Most Helpful Expert Reply
Math Expert
Joined: 02 Sep 2009
Posts: 65771
Re: How many combinations of three letters taken from letters  [#permalink]

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26 Jan 2013, 04:09
17
8
bmwhype2 wrote:
How many combinations of three letters taken from letters (a, a, b, b, c, c, d) are possible?

A. 12
B. 13
C. 35
D. 36
E. 56

This kind of question has little chances appearing on the actual test.

Anyway, we have 7 letters {a, a, b, b, c, c, d}. There are 2 ways to select 3 letters out of this set:

CASE #1: all letters are distinct:

Since there are 4 distinct letters a, b, c and d, then the # of ways to select 3 out of 4 is 4C3=4.

CASE #2: 2 letters are the same and the third is different:

There are 3 letters from the set which can provide us with two letters: a, b, and c. 3C1=3 gives the # of ways to select which letter out of these 3 will provide us with 2 letters. For, example double letters can be aa, bb, or cc.

Next, we are left with 3 letters to choose the third letter. For example, if we choose aa, then b, c, and d are left to choose from for the third letter, thus the # of ways to do that is 3C1=3.

Total # of ways for this case is therefore 3C1*3C1=9.

Total for both cases = 4+9 = 13.

Answer: B.

Hope it's clear.
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Senior Manager
Joined: 26 Jul 2007
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04 Dec 2007, 15:00
8
2
Im getting 13.

It says how many combinations. So each result must be different.

Ex. you cant use abc and cba.

You can start with the the 4 seperate letters in combination of 3.

4C3=4

You can then account for multiple letters.

Ex. aab, aac, aad,bba,bbc,bbd,cca,ccb,ccd

That's another 9 combinations.

4+9=13

Ans. C
##### General Discussion
Manager
Joined: 06 Aug 2007
Posts: 145
Re: Combinatorics - Perms  [#permalink]

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02 Dec 2007, 17:58
bmwhype2 wrote:
How many combinations of three letters taken from letters (a,a,b,b,c,c,d) are possible?
A - 12
B - 13
C - 35
D - 36
E - 56

I am getting 35-8 = 23 ..but none of the choices is that.
Manager
Joined: 03 Sep 2006
Posts: 161
Re: Combinatorics - Perms  [#permalink]

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03 Dec 2007, 00:35
spider wrote:
bmwhype2 wrote:
How many combinations of three letters taken from letters (a,a,b,b,c,c,d) are possible?
A - 12
B - 13
C - 35
D - 36
E - 56

I am getting 35-8 = 23 ..but none of the choices is that.

35 - 8 will be 27 =)

I'll bet on simplicity here 7C3 = 35
Senior Manager
Joined: 09 Oct 2007
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03 Dec 2007, 01:25
I agree on 35. Problem doesn't say letters must be different or anything else like that.
SVP
Joined: 29 Mar 2007
Posts: 1648
Re: Combinatorics - Perms  [#permalink]

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03 Dec 2007, 21:36
1
bmwhype2 wrote:
How many combinations of three letters taken from letters (a,a,b,b,c,c,d) are possible?
A - 12
B - 13
C - 35
D - 36
E - 56

7!/3!4! --> 35.

hehe 35-8=23 ---> I always make stupid errors such as this.
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Joined: 17 Nov 2007
Posts: 2913
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Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Re: Combinatorics - Perms  [#permalink]

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03 Jan 2008, 15:44
4
2
B

4C3=4 - all different letters
3C1*3C1=9 - two letters are the same.
N=4+9=13
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Joined: 18 Aug 2009
Posts: 231
Re: Combinatorics - Perms  [#permalink]

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27 Sep 2009, 21:15
3
$$4C3+3C1*3C1 = 13$$
Manager
Joined: 29 Nov 2009
Posts: 95
Location: United States
Re: Combinatorics - Perms  [#permalink]

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16 Feb 2010, 09:49
2
jeeteshsingh - you use it to account for the following:

aa w/ b,c,d (aac, aab, aad) - 3 of these
bb w/ a,c,d -3 of these
cc w/ a,b,d -3 of these

3C1 is choose 3 letters w/ two of the same letter

It makes more sense to me as 3*3C1 = 9 total (rather than 3C1*3C1)
Manager
Joined: 01 Feb 2010
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Re: Combinatorics - Perms  [#permalink]

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16 Feb 2010, 10:09
bmwhype2 wrote:
How many combinations of three letters taken from letters (a,a,b,b,c,c,d) are possible?
A - 12
B - 13
C - 35
D - 36
E - 56

7C3 = 35 hence C.

Is it really 700 plus question as mentioned in tag.
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Joined: 01 Feb 2010
Posts: 160
Re: Combinatorics - Perms  [#permalink]

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16 Feb 2010, 21:59
johnnymac wrote:
The answer is 13, not 35.

Question does not say that we cannot select duplicate letters. So to select 3 charaters from 7 characters hence 7C3.
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Re: Combinatorics - Perms  [#permalink]

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17 Feb 2010, 02:21
bangalorian2000 wrote:
johnnymac wrote:
The answer is 13, not 35.

Question does not say that we cannot select duplicate letters. So to select 3 charaters from 7 characters hence 7C3.

This isn't correct. You need to take into account that some of the elements are duplicate. With your logic.. if u have the letters as {a,a,a,a,a,a,a}... then would no of comb possible for 3 letter word be 7c3? I don't think so.

Your answer would only be correct if all the 7 letters were different. If anyone of them repeated, the combinations would become less in number. Please check!
Intern
Joined: 20 Oct 2011
Posts: 12
Re: How many combinations of three letters taken from letters  [#permalink]

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10 Aug 2012, 16:21
I always learn by modifying problems a little bit and see what answers I can come up with to test my understanding so...

If I assume that there's an extra a, an extra b, and an extra c for example. i.e. We have (a,a,a,b,b,b,c,c,c,d). Will that change our computation of 4C3+3C1*3C1? I myself don't think so but I'm waiting for your comments.

If I assume that's there's an an extra couple of D's and E's and 1 F i.e. (a,a,b,b,c,c,d,d,d,e,e,f). In this case the answer, as I guess, is 6C3+5C3*4C1=60. Am I correct?

Thanks
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Re: How many combinations of three letters taken from letters  [#permalink]

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23 Sep 2012, 07:22
Just got confused.

approaching by filling in slots.

we have three slots for seven letters.

now the first slot can be filled by any of the 7 letters, then 6, then 5.

so if there were 7 different letters, we would have had 7*6*5 and remove the repetitive combinations by dividing with 3! that makes 35 combinations.

For this approach, I got stuck here not knowing how to delete the repetitive combinations due to double letters. Help please.

P.S: I got the answer through other approach by adding unique and double letter combinations, I just want to understand why I got stuck above, rather how to continue from above.
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Re: How many combinations of three letters taken from letters  [#permalink]

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25 Jan 2013, 11:36
Hey could someone please explain how they got 3C1 3C1? There are only two elements chosen in that situation? There doesn't seem to be a good explanation, or an OA for this problem
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Re: How many combinations of three letters taken from letters  [#permalink]

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25 Jan 2013, 13:11
1
1
Combinations only... total 13 (answer B)
So 4C3 for abcd - 4 choices
3 more for aa(b,c or d)
3 more for bb(a,c or d)
3 more for cc(a,b or d)

total 13
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Re: How many combinations of three letters taken from letters  [#permalink]

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24 Sep 2014, 06:57
bmwhype2 wrote:
How many combinations of three letters taken from letters (a, a, b, b, c, c, d) are possible?

A. 12
B. 13
C. 35
D. 36
E. 56

Sol:

consider any one pair with d
one pair from 3 pairs = 3c1 *1 (AAD or BBD or CCD)
But they can interchange their position(AAD,ADA,DAA) hence 3c1*1*3!/2! = 9

also we can consider all different digits

so we have 4 distinct objects and we have to choose 3 i.e. 4c3 = 4

9+4=13
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Re: How many combinations of three letters taken from letters  [#permalink]

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06 Oct 2014, 04:44
johnnymac wrote:
jeeteshsingh - you use it to account for the following:

aa w/ b,c,d (aac, aab, aad) - 3 of these
bb w/ a,c,d -3 of these
cc w/ a,b,d -3 of these

3C1 is choose 3 letters w/ two of the same letter

It makes more sense to me as 3*3C1 = 9 total (rather than 3C1*3C1)

Indeed writing it as : 2C2*3C1*3=9 will make the most sense and clear it to almost anybody. ( 2C2 refers to the choosing of same 2 letters and then 3C1 refers to chossing any one letter from among the 3 diff types of letters left and then finally *3 gives the total such possible discrete cases.)

Hope this helps!!
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Joined: 27 Apr 2016
Posts: 4
How many combinations of three letters taken from letters  [#permalink]

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23 Jun 2016, 17:12
Bunuel wrote:
bmwhype2 wrote:
How many combinations of three letters taken from letters (a, a, b, b, c, c, d) are possible?

A. 12
B. 13
C. 35
D. 36
E. 56

This kind of question has little chances appearing on the actual test.

Anyway, we have 7 letters {a, a, b, b, c, c, d}. There are 2 ways to select 3 letters out of this set:

CASE #1: all letters are distinct:

Since there are 4 distinct letters a, b, c and d, then the # of ways to select 3 out of 4 is 4C3=4.

CASE #2: 2 letters are the same and the third is different:

There are 3 letters from the set which can provide us with two letters: a, b, and c. 3C1=3 gives the # of ways to select which letter out of these 3 will provide us with 2 letters. For, example double letters can be aa, bb, or cc.

Next, we are left with 3 letters to choose the third letter. For example, if we choose aa, then b, c, and d are left to choose from for the third letter, thus the # of ways to do that is 3C1=3.

Total # of ways for this case is therefore 3C1*3C1=9.

Total for both cases = 4+9 = 13.

Answer: B.

Hope it's clear.

Hi Bunnel ,
Let -->s (same letter)
--> d(distinct letter)
Then
In case 2(two same other distinct) shouldn't we consider these cases?
-->ssd >>3*3=9
-->sds >>3*3=9
--> dss >>3*3=9
so # of possibilities for case two =9*3=27
How many combinations of three letters taken from letters   [#permalink] 23 Jun 2016, 17:12

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# How many combinations of three letters taken from letters

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