How many different arrangements can you make with the letters of the word communication if all the vowels always come together?
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Material based on Class 12th Syllabus, 10000+ Question Bank, Unlimited Chapter-Wise and Subject-Wise Mock Tests, Study Improvement Plan. ₹ 7999/- ₹ 4999/- - AI Coach Study Modules, - Unlimited Mock Tests, - Expert Mentorship, - Study Improvement Plan. ₹
9999/- ₹ 8499/- - AI Coach Study Modules, - Unlimited Mock Tests, - Expert Mentorship, - Study Improvement Plan. ₹ 13999/- ₹ 12499/- - AI Coach Study Modules, - Unlimited Mock Tests, - Expert Mentorship, - Study Improvement Plan. ₹ 9999/- ₹ 8499/- $\begingroup$ Using all the letters of the word ARRANGEMENT how many different words using all letters at a time can be made such that both A, both E, both R both N occur together .
asked Nov 13, 2012 at 15:33
$\endgroup$ 4 $\begingroup$ "ARRANGEMENT" is an eleven-letter word. If there were no repeating letters, the answer would simply be $11!=39916800$. However, since there are repeating letters, we have to divide to remove the duplicates accordingly. There are 2 As, 2 Rs, 2 Ns, 2 Es Therefore, there are $\frac{11!}{2!\cdot2!\cdot2!\cdot2!}=2494800$ ways of arranging it. answered Nov 13, 2012 at 15:43
JTJMJTJM 3871 gold badge2 silver badges12 bronze badges $\endgroup$ 2 $\begingroup$ The word ARRANGEMENT has $11$ letters, not all of them distinct. Imagine that they are written on little Scrabble squares. And suppose we have $11$ consecutive slots into which to put these squares. There are $\dbinom{11}{2}$ ways to choose the slots where the two A's will go. For each of these ways, there are $\dbinom{9}{2}$ ways to decide where the two R's will go. For every decision about the A's and R's, there are $\dbinom{7}{2}$ ways to decide where the N's will go. Similarly, there are now $\dbinom{5}{2}$ ways to decide where the E's will go. That leaves $3$ gaps, and $3$ singleton letters, which can be arranged in $3!$ ways, for a total of $$\binom{11}{2}\binom{9}{2}\binom{7}{2}\binom{5}{2}3!.$$ answered Nov 13, 2012 at 15:41
André NicolasAndré Nicolas 493k44 gold badges522 silver badges954 bronze badges $\endgroup$ 2 Math Expert Joined: 02 Sep 2009 Posts: 86773 In how many different ways can the letters of the word MISSISSIPPI be [#permalink]
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 Hide Show timer StatisticsIn how many different ways can the letters of the word MISSISSIPPI be arranged if the vowels must always be together? A. 48 _________________ GMAT Club Legend Joined: 11 Sep 2015 Posts: 6801 Location: Canada
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Bunuel wrote: In how many different ways can the letters of the word MISSISSIPPI be arranged if the vowels must always be together? A. 48 When we want to arrange a group of items in which some of the items are identical, we can use something called the MISSISSIPPI rule. It goes like this: If there are n objects where A of them are alike, another B of them are alike, another C of them are alike, and so on, then the total number of possible arrangements = n!/[(A!)(B!)(C!)....] So, for example, we can calculate the number of arrangements of the letters in MISSISSIPPI as follows: -------NOW ONTO THE QUESTION!!!----------------- First "glue" the 4 I's together to create ONE character: IIII (this ensures that they stay together) Answer: RELATED VIDEO _________________ Brent Hanneson – Creator of gmatprepnow.com Manhattan Prep Instructor Joined: 04 Dec 2015 Posts: 941 GMAT 1: 790 Q51 V49 Re: In how many different ways can the letters of the word MISSISSIPPI be [#permalink]
Bunuel wrote: In how many different ways can the letters of the word MISSISSIPPI be arranged if the vowels must always be together? A. 48 Ooh, a combinatorics problem with some fairly large numbers. The best technique for these is to mentally break them down into simpler problems. Otherwise, the solutions tend to sort of look like magic tricks - sure, you can use the formula, but how are you supposed to know to use that formula, and not one of the many similar-looking ones? There are four vowels, and they're all the same. Let's set those vowels aside: (IIII) Now we have the letters MSSSSPP. How many ways can just those letters be arranged? Well, if they were all different, we could arrange them in 7x6x5x4x3x2x1 = 7! ways. However, they aren't all different. For instance, four of the letters are (S). I'm going to color them to demonstrate why that matters: one arrangement: MSSSSPP We counted both of those arrangements, but we actually don't want to. All of the Ss look the same - they aren't actually different colors - so we want to make those arrangements the same. Because there are 4x3x2x1 = 24 ways to order the different Ss, we want every set of 24 arrangements where the Ss are in the same place, to just count as 1 arrangement, instead. So, we can divide out the extra possibilities by dividing our total by 24 (or 4!): 7!/4! Do the same thing with the two Ps. We still have twice as many arrangements as we need, since we've counted as if the two Ps were different, but they're actually the same. So, divide the total by 2: 7!/(4! x 2) Now we have to put the (IIII) letters back in. They all have to go together. Start by looking at one of the arrangements of the other letters: PMSSPSS. Where can the four Is go? There are 8 places where we can put them: IIIIPMSSPSS etc. So, for each arrangement, we have to multiply by 8, to account for the eight possible ways to put the vowels back in. Here's the final answer: (8 x 7!) / (4! x 2) = (8 x 7 x 6 x 5 x 4 x 3 x 2) / (4 x 3 x 2 x 2) = 4 x 7 x 6 x 5 = 4 x 210 = 840. Verbal Forum Moderator Joined: 08 Dec 2013 Status:Greatness begins beyond your comfort zone Posts: 2214 Location: India Concentration: General Management, Strategy GPA: 3.2 WE:Information Technology (Consulting)
Re: In how many different ways can the letters of the word MISSISSIPPI be [#permalink]
The word MISSISSIPPI has 4 I , 4S , 2P and M . Number of different ways = 8!/(4!*2!) When everything seems to be going against you, remember that the airplane takes off against the wind, not with it. - Henry Ford Director Joined: 05 Mar 2015 Posts: 907
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Bunuel wrote: In how many different ways can the letters of the word MISSISSIPPI be arranged if the vowels must always be together? A. 48 vowel (I) together can be arranged with other 7 letters in 8! ways different ways = 8!*4!/( 4!*4!*2!) = 840---(4I's , 4 S's , 2 P's) Ans E Intern Joined: 17 Nov 2016 Posts: 23 Re: In how many different ways can the letters of the word MISSISSIPPI be [#permalink]
What about this approach? Stage 1: Lets place the letters without the 4I's: Stage
2: Now lets place the 4I's Total ways= 105*8= 840 ways. Is this a correct approach? Intern Joined: 25 May 2017 Posts: 10 Re: In how many different ways can the letters of the word MISSISSIPPI be [#permalink]
Rearranging letters has a simple formula: Total number of letters Mississippi has: 4 s's 4 i's 2 p's Total number of letters: 11! So the answer is \(\frac{11!}{4!4!2!}\) GmatPrepLondon Visit: http://www.gmatpreplondon.co.uk Material offered free of charge along with full end-to-end GMAT Quantitative Course: UNC Kenan Flagler Moderator Joined: 18 Jul 2015 Posts: 248 GMAT 1: 530 Q43 V20 WE:Analyst (Consumer Products)
In how many different ways can the letters of the word MISSISSIPPI be [#permalink]
GmatIvyPrep wrote: Rearranging letters has a simple formula: Total number of letters
Mississippi has: 4 s's 4 i's 2 p's Total number of letters: 11! So the answer is \(\frac{11!}{4!4!2!}\) Hi GmatIvyPrep This does not seem to be the correct answer as you may have overlooked the
constraint of keeping all vowels (letter I in this case) together. Warm Regards, Cheers. Wishing Luck to Every GMAT Aspirant | Press +1 if this post helped you! Interested in data analysis & reporting using R programming? - https://www.youtube.com/watch?v=ZOJHBYhmD2I GMAT Club Legend Joined: 03 Jun 2019 Posts: 4773 Location: India GMAT 1: 690 Q50 V34 WE:Engineering (Transportation)
Re: In how many different ways can the letters of the word MISSISSIPPI be
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Bunuel wrote: In how many different ways can the letters of the word MISSISSIPPI be arranged if the vowels must always be together? A. 48 Asked: In how many different ways can the letters of the word MISSISSIPPI be arranged if the vowels must always be together? 1. M-1 Vowels = I 1. M-1 No of arrangements = 8! / (4! * 2!) = 840 ways IMO E
Kinshook Chaturvedi Non-Human User Joined: 09 Sep 2013 Posts: 24373 Re: In how many different ways can the letters of the word MISSISSIPPI be
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Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. Re: In how many different ways can the letters of the word MISSISSIPPI be [#permalink] 13 Oct 2021, 07:06 Moderators: Senior Moderator - Masters Forum 3084 posts How many different ways can the letters of the word always come together?The number of ways the word TRAINER can be arranged so that the vowels always come together are 360.
How many words can be formed combination so that the vowels always come together?Hence, the answer is 462.
How many of them have all vowels occur together?Total numbers of letters, we have to arrange = 5+1=6. Total number of arrangements when all vowels occur together=720×6=4320.
How many ways can the letters of the word ideal be arranged so that all the consonants are always together?∴ The number of ways of arranging these letter is 64800.
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