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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 ## In how many ways can the letters of word "EDUCATION&quo tagged by: VJesus12 ##### This topic has 4 expert replies and 0 member replies ## In how many ways can the letters of word "EDUCATION&quo In how many ways can the letters of word "EDUCATION" be arranged such that all the vowels, as well as all the Consonants, always appear together? A) 9! B) 5!*4! C) 5!*5! D) 5!*4!*2! E) 6!*4! The OA is D. Why is D? I'm confused here. Experts, may you give me some help? Please. ### GMAT/MBA Expert GMAT Instructor Joined 04 Oct 2017 Posted: 551 messages Followed by: 11 members Upvotes: 180 Top Reply VJesus12 wrote: In how many ways can the letters of word "EDUCATION" be arranged such that all the vowels, as well as all the Consonants, always appear together? A) 9! B) 5!*4! C) 5!*5! D) 5!*4!*2! E) 6!*4! The OA is D. Why is D? I'm confused here. Experts, may you give me some help? Please. Hello Vjesus12. Let's take a look at the question. EDUCATION has 5 vowels and 4 consonants. Hence, there are 5! ways of arranged the vowels and there are 4! ways to arrange the consonants. Now, let's call A to all the vowels and B all the consonants together. Hence, we have two ways to arrange them, that is to say, AB and BA. Finally, the total ways of arrange the vowels and consonants as asked are 5!*4!*2 = 5!*4!*2!. Therefore, the correct answer is the option D. I hope this answer may help you. Feel free to ask me again if you have a doubt. Regards. _________________ GMAT Prep From The Economist We offer 70+ point score improvement money back guarantee. Our average student improves 98 points. Free 7-Day Test Prep with Economist GMAT Tutor - Receive free access to the top-rated GMAT prep course including a 1-on-1 strategy session, 2 full-length tests, and 5 ask-a-tutor messages. Get started now. ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 13043 messages Followed by: 1251 members Upvotes: 5254 GMAT Score: 770 VJesus12 wrote: In how many ways can the letters of word "EDUCATION" be arranged such that all the vowels, as well as all the Consonants, always appear together? A) 9! B) 5!*4! C) 5!*5! D) 5!*4!*2! E) 6!*4! VOWELS: E, U, A, I, O CONSONANTS: D, C, T, N Take the task of arranging the letters and break it into stages. Stage 1: Arrange the 5 vowels We can arrange n unique objects in n! ways So, we can arrange the 5 vowels in 5! ways In other words, we can complete stage 1 in 5! ways Stage 2: Arrange the 4 consonants There are 4 unique objects to arrange. We can complete stage 2 in 4! ways Stage 3: The two groups of letters (the vowel group and the consonant group) There are two options: 1) CONSONANTS-VOWELS 1) VOWELS-CONSONANTS We can complete stage 3 in 2 ways By the Fundamental Counting Principle (FCP), we can complete the 3 stages (and thus arrange all 9 letters) in (5!)(4!)(2) ways. Answer: D -------------------------- Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch our free video: http://www.gmatprepnow.com/module/gmat-counting/video/775 You can also watch a demonstration of the FCP in action: https://www.gmatprepnow.com/module/gmat-counting/video/776 Then you can try solving the following questions: EASY - http://www.beatthegmat.com/what-should-be-the-answer-t267256.html - http://www.beatthegmat.com/counting-problem-company-recruitment-t244302.html - http://www.beatthegmat.com/picking-a-5-digit-code-with-an-odd-middle-digit-t273110.html - http://www.beatthegmat.com/permutation-combination-simple-one-t257412.html - http://www.beatthegmat.com/simple-one-t270061.html MEDIUM - http://www.beatthegmat.com/combinatorics-solution-explanation-t273194.html - http://www.beatthegmat.com/arabian-horses-good-one-t150703.html - http://www.beatthegmat.com/sub-sets-probability-t273337.html - http://www.beatthegmat.com/combinatorics-problem-t273180.html - http://www.beatthegmat.com/digits-numbers-t270127.html - http://www.beatthegmat.com/doubt-on-separator-method-t271047.html - http://www.beatthegmat.com/combinatorics-problem-t267079.html DIFFICULT - http://www.beatthegmat.com/wonderful-p-c-ques-t271001.html - http://www.beatthegmat.com/permutation-and-combination-t273915.html - http://www.beatthegmat.com/permutation-t122873.html - http://www.beatthegmat.com/no-two-ladies-sit-together-t275661.html - http://www.beatthegmat.com/combinations-t123249.html 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! ### GMAT/MBA Expert GMAT Instructor Joined 25 May 2010 Posted: 15387 messages Followed by: 1872 members Upvotes: 13060 GMAT Score: 790 VJesus12 wrote: In how many ways can the letters of word "EDUCATION" be arranged such that all the vowels, as well as all the Consonants, always appear together? A) 9! B) 5!*4! C) 5!*5! D) 5!*4!*2! E) 6!*4!. Number of ways to arrange the 5 vowels E, A, U, I, and O = 5!. Number of ways to arrange the 4 consonants D, C, T and N = 4!. To combine the options above, we multiply: 5!4!. Since VOWELS-CONSONANTS and CONSONANTS-VOWELS both constitute viable arrangements, the result above must be multiplied by 2: 5!4!2. The correct answer is D. _________________ Mitch Hunt Private Tutor for the GMAT and GRE GMATGuruNY@gmail.com If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon. Available for tutoring in NYC and long-distance. For more information, please email me at GMATGuruNY@gmail.com. Student Review #1 Student Review #2 Student Review #3 Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now. ### GMAT/MBA Expert GMAT Instructor Joined 09 Apr 2015 Posted: 1465 messages Followed by: 19 members Upvotes: 39 VJesus12 wrote: In how many ways can the letters of word "EDUCATION" be arranged such that all the vowels, as well as all the Consonants, always appear together? A) 9! B) 5!*4! C) 5!*5! D) 5!*4!*2! E) 6!*4! We need to determine the number of ways to arrange the letters of word "EDUCATION" such that all the vowels as well as all the consonants always appear together. We start by arranging the given word with vowels together with the consonants together. [EUAIO] [DCTN] We first see that there are 2 ways to arrange [EUAIO] and [DCTN]: [EUAIO] [DCTN] and [DCTN] [EUAIO]. Next we must determine the number of arrangements for the letters in [EUAIO] and in [DCTN]. [EUAIO] can be arranged in 5! ways. [DCTN] can be arranged in 4! ways. Thus, the total number of ways to arrange the letters of word "EDUCATION" such that all the vowels as well as all the consonants always appear together is 5! x 4! x 2, or 5! x 4! x 2!. Answer: D _________________ Jeffrey Miller Head of GMAT Instruction jeff@targettestprep.com See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. 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