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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## Combinatorics: Solution Explanation tagged by: Brent@GMATPrepNow ##### This topic has 4 expert replies and 5 member replies ## Combinatorics: Solution Explanation 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 ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 13043 messages Followed by: 1251 members Upvotes: 5254 GMAT Score: 770 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 ( = D) Cheers, Brent Aside: For more information about the FCP, watch our free video: http://www.gmatprepnow.com/module/gmat-counting?id=775 _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! Senior | Next Rank: 100 Posts Joined 03 Mar 2011 Posted: 40 messages Followed by: 7 members Upvotes: 3 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, Rahul Sehgal ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 13043 messages Followed by: 1251 members Upvotes: 5254 GMAT Score: 770 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 Cheers, Brent _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! Senior | Next Rank: 100 Posts Joined 03 Mar 2011 Posted: 40 messages Followed by: 7 members Upvotes: 3 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 Master | Next Rank: 500 Posts Joined 07 Mar 2015 Posted: 111 messages Followed by: 1 members Upvotes: 8 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. ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 13043 messages Followed by: 1251 members Upvotes: 5254 GMAT Score: 770 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, Brent _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! Master | Next Rank: 500 Posts Joined 07 Mar 2015 Posted: 111 messages Followed by: 1 members Upvotes: 8 Thanks Brent, I think I should better work on my speed. ~Binit. ### GMAT/MBA Expert GMAT Instructor Joined 12 Sep 2012 Posted: 2635 messages Followed by: 117 members Upvotes: 625 Target GMAT Score: V51 GMAT Score: 780 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. Enroll in a Veritas Prep GMAT class completely for FREE. Wondering if a GMAT course is right for you? Attend the first class session of an actual GMAT course, either in-person or live online, and see for yourself why so many students choose to work with Veritas Prep. Find a class now! Master | Next Rank: 500 Posts Joined 07 Mar 2015 Posted: 111 messages Followed by: 1 members Upvotes: 8 Quote: 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. • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • Award-winning private GMAT tutoring Register now and save up to$200

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