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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).

3. aabc - from 4 letters 2 are the same and other 2 are different: 3C1*7C1*6C1*\frac{4!}{2!}=756 - 3C1 choosing which letter will proved with 2 letters (out of 3 double letter - MAT), 7C1 choosing third letter out of 7 distinct letters left, 6C1 choosing fourth letter out of 6 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).

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

Y is it 7C1*6C1? selecting 2 from 7 is 7C2?....
_________________

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

3. aabc - from 4 letters 2 are the same and other 2 are different: 3C1*7C1*6C1*\frac{4!}{2!}=756 - 3C1 choosing which letter will proved with 2 letters (out of 3 double letter - MAT), 7C1 choosing third letter out of 7 distinct letters left, 6C1 choosing fourth letter out of 6 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).

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

Y is it 7C1*6C1? selecting 2 from 7 is 7C2?....

It's a typo. There should be 7C1*6C1/2, which is in fact 7C2. Edited.
_________________

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!

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).

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

Answer: 2454.

Hi Bunnel, Is this a GMAT worthy question?

No, but this question is good to practice.
_________________

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

Answer: 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?

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

Answer: 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.