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Combinatorics: Solution Explanation

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Combinatorics: Solution Explanation

by Dblooos » Sat Jan 04, 2014 9:36 pm
Find the number of ways the letters of the word "MACHINE" can be arranged so that vowels may occupy only odd positions?

A. 3! X 4!
B. 7P3 X 4!
c. 7P4 X 3!
D. None of these


Here is how I approached it:

There are three vowels (A, I, E)and four consonants (M, C, H, N). So I treat it just like a lock code problem.

- - - - - - -
1 2 3 4 5 6 7

So Letter A has 4 choices (1, 3, 5, 7)
Letter I has 3 choices (one less than letter A)
Letter E has 2 choices.

Once these three are placed.

There are 4 places remaining and 4 consonants. Which can be arranged in 4! ways.

So the total number of ways, it can be done is

4 x 3 x 2 x 4!.

or 4! X 4!

Am I correct or am I double counting somewhere? Please advise. Thank you

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by Brent@GMATPrepNow » Sun Jan 05, 2014 8:04 am
Dblooos wrote:Find the number of ways the letters of the word "MACHINE" can be arranged so that vowels may occupy only odd positions?

A. 3! X 4!
B. 7P3 X 4!
c. 7P4 X 3!
D. None of these
Your solution looks great, Dbloos. Here's my solution with some rationale added.


Take the task of building words and break it into stages.

We have 7 spaces for letters: _ _ _ _ _ _ _

We'll begin with the most RESTRICTIVE stages (placing the vowels)

Stage 1: Place the A
The A can go in spaces #1, #3, #5 or #7, so we can complete stage 1 in 4 ways

Stage 2: Place the I
Now that we've placed the A, there are 3 acceptable spaces remaining, so we complete this stage in 3 ways.

Stage 3: Place the E
Now that we've placed the A and I, there are 2 acceptable spaces remaining, so we complete this stage in 2 ways.

Stage 4: Place the M
There are 4 spaces remaining, so we complete this stage in 4 ways.

Stage 5: Place the C
There are 3 spaces remaining, so we complete this stage in 3 ways.

Stage 6: Place the H
We complete this stage in 2 ways.

Stage 7: Place the N
We complete this stage in 1 way.

By the Fundamental Counting Principle (FCP), we can complete all 7 stages (and thus build 7-letter words) in (4)(3)(2)(4)(3)(2)(1) ways ([spoiler] = D[/spoiler])

Cheers,
Brent

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by rahul.sehgal@btgchampion » Sat Feb 01, 2014 6:15 am
Can we start with a vowel other than 'A' in the first place ?

I believe, we can as we do not need to worry about the order of vowels here. It is just that they had to occupy odd positions ? Am I missing something in my understanding ?
Best Regards,
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by Brent@GMATPrepNow » Sat Feb 01, 2014 8:06 am
rahul.sehgal@mbawatch wrote:Can we start with a vowel other than 'A' in the first place ?

I believe, we can as we do not need to worry about the order of vowels here. It is just that they had to occupy odd positions ? Am I missing something in my understanding ?
Sure, stage 1 can be selecting a place for ANY vowel. I just chose A because it was the first that came to mind.

If you find places for E, then A, and then I, the answer will still be D

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by rahul.sehgal@btgchampion » Sat Feb 01, 2014 11:58 am
Thank you Brent !!

I have a very small confusion regarding FCP and combinations as I have recently started working on combinatorics. It is like - are these used independent of each other ? I understand the combinatorics become permutation when order matters and combinations when it does not.

I mean, combination is basically counting and FCP lays the foundation for it. Can you provide an example for me that can help me differentiate between the two. I sometimes get confuse among the two. Currently, While solving questions on these, I am not able to make up sometimes whether I can use FCP or not and if yes, how can I go about dividing them into stages. May be this is something, that will come with practice. Thanks in advance for your responses !!
Best Regards,
Rahul Sehgal

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by binit » Mon Apr 27, 2015 9:54 pm
Hi Brent,
I am happy that I am able to solve it correctly at first attempt, though it took me 4 min. Your method is far more easier, thanks a lot for that.
Do u think this type of problems could be seen in the real test?

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by Brent@GMATPrepNow » Tue Apr 28, 2015 9:44 am
binit wrote:Hi Brent,
I am happy that I am able to solve it correctly at first attempt, though it took me 4 min. Your method is far more easier, thanks a lot for that.
Do u think this type of problems could be seen in the real test?

~Binit.
Hi Binit,

I think the skills tested here are within the scope of the GMAT.
I'd say it's a 650+ question.

Cheers,
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by binit » Sat May 02, 2015 6:16 am
Thanks Brent,
I think I should better work on my speed.

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by Matt@VeritasPrep » Sun May 03, 2015 10:44 pm
binit wrote:Thanks Brent,
I think I should better work on my speed.

~Binit.


IMO speed is mostly a byproduct of conceptual understanding: after all, understanding will help you go faster, but going faster usually does NOT help you understand! Work on your problem solving skills and pick up new ones from good explanations, and before you know it you'll be blazing through these problems.

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by binit » Mon May 04, 2015 11:22 pm
IMO speed is mostly a byproduct of conceptual understanding: after all, understanding will help you go faster, but going faster usually does NOT help you understand! Work on your problem solving skills and pick up new ones from good explanations, and before you know it you'll be blazing through these problems.
Thanks a lot Matt, for the insightful suggestion. I ll follow that actively.

~Binit.