# How many different words can be formed by taking 4 letters out of the letters of word again with each word has to start with a?

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How many words can be formed by taking 4 letters at a time out of the [#permalink]

Updated on: 26 Feb 2019, 03:23

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How many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.

A. 756
B. 1680
C. 1698
D. 2436
E. 2454

Originally posted by jatt86 on 14 Apr 2010, 04:33.
Last edited by Bunuel on 26 Feb 2019, 03:23, edited 1 time in total.

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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

14 Apr 2010, 05:45

jatt86 wrote:

1) how many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.

There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice (double letter).

Selected 4 letters can have following 3 patterns:

1. abcd - all 4 letters are different:
$$8P4=1680$$ (choosing 4 distinct letters out of 8, when order matters) or $$8C4*4!=1680$$ (choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct letters);

2. aabb - from 4 letters 2 are the same and other 2 are also the same:
$$3C2*\frac{4!}{2!2!}=18$$ - 3C2 choosing which two double letter will provide two letters (out of 3 double letter - MAT), multiplying by $$\frac{4!}{2!2!}$$ to get different arrangements (for example MMAA can be arranged in $$\frac{4!}{2!2!}$$ # of ways);

3. aabc - from 4 letters 2 are the same and other 2 are different:
$$3C1*7C2*\frac{4!}{2!}=756$$ - 3C1 choosing which letter will proved with 2 letters (out of 3 double letter - MAT), 7C2 choosing third and fourth letters out of 7 distinct letters left and multiplying by $$\frac{4!}{2!}$$ to get different arrangements (for example MMIC can be arranged in $$\frac{4!}{2!}$$ # of ways).

1680+18+756=2454

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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

14 Apr 2010, 06:40

a very similar question:
Find the no: of 4 letter words that can be formed from the string "AABBBBCC" ?

Here we have 3 distinct letters(A,B,C) & 4 slots to fill. What logic do you use to solve this problem?

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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

14 Apr 2010, 07:01

idiot wrote:

a very similar question:
Find the no: of 4 letter words that can be formed from the string "AABBBBCC" ?

Here we have 3 distinct letters(A,B,C) & 4 slots to fill. What logic do you use to solve this problem?

Three patterns:

1. XXXX - only BBBB, so 1
2. XXYY - 3C2(choosing which will take the places of X and Y from A, B and C)*4!/2!2!(arranging)=18
3. XXYZ - 3C1(choosing which will take the place of X from A, B and C)*4!/2!(arranging)=36
4. XXXY - 2C1(choosing which will take the place of Y from A and C, as X can be only B)*4!/3!(arranging)=8

1+18+36+8=63
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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

14 Apr 2010, 10:41

thanks a ton, bunuel

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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

15 Apr 2010, 06:20

I'm usually not bad with anagram problems like this but the term "words" threw me off completely.
For some reason I assumed the combination of letters had to combine to make sense, i.e. a "word".

MTHE - is hardly a word, so i started counting actual "words"... so obviously completely bombed the question!

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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

25 May 2013, 17:50

Bunuel wrote:

jatt86 wrote:

1) how many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.

There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice (double letter).

Selected 4 letters can have following 3 patterns:

1. abcd - all 4 letters are different:
$$8P4=1680$$ (choosing 4 distinct letters out of 8, when order matters) or $$8C4*4!=1680$$ (choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct letters);

2. aabb - from 4 letters 2 are the same and other 2 are also the same:
$$3C2*\frac{4!}{2!2!}=18$$ - 3C2 choosing which two double letter will provide two letters (out of 3 double letter - MAT), multiplying by $$\frac{4!}{2!2!}$$ to get different arrangements (for example MMAA can be arranged in $$\frac{4!}{2!2!}$$ # of ways);

3. aabc - from 4 letters 2 are the same and other 2 are different:
$$3C1*7C2*\frac{4!}{2!}=756$$ - 3C1 choosing which letter will proved with 2 letters (out of 3 double letter - MAT), 7C2 choosing third and fourth letters out of 7 distinct letters left and multiplying by $$\frac{4!}{2!}$$ to get different arrangements (for example MMIC can be arranged in $$\frac{4!}{2!}$$ # of ways).

1680+18+756=2454

Hi Bunnel,
Is this a GMAT worthy question?

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Posts: 86800

Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

26 May 2013, 03:13

cumulonimbus wrote:

Bunuel wrote:

jatt86 wrote:

1) how many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.

There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice (double letter).

Selected 4 letters can have following 3 patterns:

1. abcd - all 4 letters are different:
$$8P4=1680$$ (choosing 4 distinct letters out of 8, when order matters) or $$8C4*4!=1680$$ (choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct letters);

2. aabb - from 4 letters 2 are the same and other 2 are also the same:
$$3C2*\frac{4!}{2!2!}=18$$ - 3C2 choosing which two double letter will provide two letters (out of 3 double letter - MAT), multiplying by $$\frac{4!}{2!2!}$$ to get different arrangements (for example MMAA can be arranged in $$\frac{4!}{2!2!}$$ # of ways);

3. aabc - from 4 letters 2 are the same and other 2 are different:
$$3C1*7C2*\frac{4!}{2!}=756$$ - 3C1 choosing which letter will proved with 2 letters (out of 3 double letter - MAT), 7C2 choosing third and fourth letters out of 7 distinct letters left and multiplying by $$\frac{4!}{2!}$$ to get different arrangements (for example MMIC can be arranged in $$\frac{4!}{2!}$$ # of ways).

1680+18+756=2454

Hi Bunnel,
Is this a GMAT worthy question?

No, but this question is good to practice.
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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

02 Jul 2013, 10:45

Bunuel wrote:

jatt86 wrote:

1) how many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.

There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice (double letter).

Selected 4 letters can have following 3 patterns:

1. abcd - all 4 letters are different:
$$8P4=1680$$ (choosing 4 distinct letters out of 8, when order matters) or $$8C4*4!=1680$$ (choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct letters);

2. aabb - from 4 letters 2 are the same and other 2 are also the same:
$$3C2*\frac{4!}{2!2!}=18$$ - 3C2 choosing which two double letter will provide two letters (out of 3 double letter - MAT), multiplying by $$\frac{4!}{2!2!}$$ to get different arrangements (for example MMAA can be arranged in $$\frac{4!}{2!2!}$$ # of ways);

3. aabc - from 4 letters 2 are the same and other 2 are different:
$$3C1*7C2*\frac{4!}{2!}=756$$ - 3C1 choosing which letter will proved with 2 letters (out of 3 double letter - MAT), 7C2 choosing third and fourth letters out of 7 distinct letters left and multiplying by $$\frac{4!}{2!}$$ to get different arrangements (for example MMIC can be arranged in $$\frac{4!}{2!}$$ # of ways).

1680+18+756=2454

Bunuel, this is a damn hard question and I find myself not fully able to understand your logic. I am from a very weak background but I have poured through all of the MGMAT math books (excluding the advanced one) several times and still find myself unable to intutively figure out the steps to this problem.

What extra review would you suggest so I can be able to at least follow your solutions to these answers?

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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

02 Jul 2013, 11:21

tmipanthers wrote:

Bunuel wrote:

jatt86 wrote:

1) how many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.

There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice (double letter).

Selected 4 letters can have following 3 patterns:

1. abcd - all 4 letters are different:
$$8P4=1680$$ (choosing 4 distinct letters out of 8, when order matters) or $$8C4*4!=1680$$ (choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct letters);

2. aabb - from 4 letters 2 are the same and other 2 are also the same:
$$3C2*\frac{4!}{2!2!}=18$$ - 3C2 choosing which two double letter will provide two letters (out of 3 double letter - MAT), multiplying by $$\frac{4!}{2!2!}$$ to get different arrangements (for example MMAA can be arranged in $$\frac{4!}{2!2!}$$ # of ways);

3. aabc - from 4 letters 2 are the same and other 2 are different:
$$3C1*7C2*\frac{4!}{2!}=756$$ - 3C1 choosing which letter will proved with 2 letters (out of 3 double letter - MAT), 7C2 choosing third and fourth letters out of 7 distinct letters left and multiplying by $$\frac{4!}{2!}$$ to get different arrangements (for example MMIC can be arranged in $$\frac{4!}{2!}$$ # of ways).

1680+18+756=2454

Bunuel, this is a damn hard question and I find myself not fully able to understand your logic. I am from a very weak background but I have poured through all of the MGMAT math books (excluding the advanced one) several times and still find myself unable to intutively figure out the steps to this problem.

What extra review would you suggest so I can be able to at least follow your solutions to these answers?

This question is out of the scope of the GMAT, so I wouldn't worry about it too much.

As for the recommendations.

Best GMAT Books: best-gmat-math-prep-books-reviews-recommendations-77291.html

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

Hope it helps.
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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

24 Mar 2014, 12:36

If this would have been a word with three of the same letter I'm assuming you would have more than 3 combinations?

Thanks!

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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

25 Mar 2014, 01:43

Mbearmann wrote:

If this would have been a word with three of the same letter I'm assuming you would have more than 3 combinations?

Thanks!

Yes, we would have one more combination {a, a, a, b}.
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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

10 Jul 2015, 10:44

I can't understand this question. Why is this a combination question and not permutation? isnt it asking for arrangements?

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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

10 Jul 2015, 11:03

ddg wrote:

I can't understand this question. Why is this a combination question and not permutation? isnt it asking for arrangements?

You are half right.

Permutations = Combinations * n! (where n is the number of 'elements'). In this question, you first need to select the letters out of the given one (combination implied as selection = combination!!) and only after you have selected the letters , you can look at the arrangements. You can not directly go to arrangements as you need to follow the 2 step process:

1. Choose 4 out of 11 letters
2. Arrangement of those selections of 4 letters to get all the possible arrangements.

Your approach would have been correct, had the question ask us to arrange all of these 11 letters into words of 11 letters or if all the letters were different.
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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

10 Jul 2015, 12:24

Thanks!

this helped

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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

17 Aug 2019, 23:09

jatt86 wrote:

How many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.

A. 756
B. 1680
C. 1698
D. 2436
E. 2454

Asked: How many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.

M-2
A-2
T-2
H-1
E-1
I-1
C-1
S-1

Words of the form abcd = $$^8C_4 * 4! = 1680$$

Words of the form aabc = $$^3C_1*^7C_2* 4!/2! = 3*21*12 = 756$$

Words of the form aabb = $$^3C_2 * 4!/2!/2! = 3* 24/4 = 18$$

Number of words that can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS = 1680+756+18=2454

IMO E
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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

24 Feb 2020, 15:39

I understand the solution, but where does this logic break down?
MATHEMATICS = 11 letters
MM AA TT = 3 groups of 2 repeats

(11*10*9*8)/(2!2!2!) = 990

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Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

27 Feb 2020, 17:50

Bunuel wrote:

jatt86 wrote:

1) how many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.

There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice (double letter).

Selected 4 letters can have following 3 patterns:

1. abcd - all 4 letters are different:
$$8P4=1680$$ (choosing 4 distinct letters out of 8, when order matters) or $$8C4*4!=1680$$ (choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct letters);

2. aabb - from 4 letters 2 are the same and other 2 are also the same:
$$3C2*\frac{4!}{2!2!}=18$$ - 3C2 choosing which two double letter will provide two letters (out of 3 double letter - MAT), multiplying by $$\frac{4!}{2!2!}$$ to get different arrangements (for example MMAA can be arranged in $$\frac{4!}{2!2!}$$ # of ways);

3. aabc - from 4 letters 2 are the same and other 2 are different:
$$3C1*7C2*\frac{4!}{2!}=756$$ - 3C1 choosing which letter will proved with 2 letters (out of 3 double letter - MAT), 7C2 choosing third and fourth letters out of 7 distinct letters left and multiplying by $$\frac{4!}{2!}$$ to get different arrangements (for example MMIC can be arranged in $$\frac{4!}{2!}$$ # of ways).

1680+18+756=2454

hey! Why are we taking 4C2 in the aabb combination? If we are looking to calculate how the letters are arranged, shouldn't we be using 4P2 instead?

Thanks

Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]

27 Feb 2020, 17:50

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