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In how many different ways can the letters of the word MISSISSIPPI be [#permalink]
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Question Stats: 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 B. 144 C. 210 D. 420 E. 840 _________________
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 B. 144 C. 210 D. 420 E. 840 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:There are 11 letters in total There are 4 identical I's There are 4 identical S's There are 2 identical P's So, the total number of possible arrangements = 11!/[(4!)(4!)(2!)]-------NOW ONTO THE QUESTION!!!----------------- First "glue" the 4 I's together to create ONE character: IIII (this ensures that they stay together) So, basically, we must determine the number of arrangements of M, S, S, S, S, P, P and IIIIThere are 8 characters in total There are 4 identical S's There are 2 identical P's So, the total number of possible arrangements = 8!/[(4!)(2!)]= 840 Answer: RELATED VIDEO_________________
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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 B. 144 C. 210 D. 420 E. 840 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 a 'different' arrangment: MSSSSPPWe 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:IIIIPMSSPSSPIIIIMSSPSSPMIIIISSPSSPMSIIIISPSSetc. 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. _________________
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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 .The only vowel present in the word is I and we have to consider 4 I's as a single groupNumber of different ways = 8!/(4!*2!)=(8*7*6*5)/2=840Answer E _________________
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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 B. 144 C. 210 D. 420 E. 840 vowel (I) together can be arranged with other 7 letters in 8! ways4 no. vowels themselves can be arranged in 4! waysdifferent ways = 8!*4!/( 4!*4!*2!) = 840---(4I's , 4 S's , 2 P's)Ans E
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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:MSSSSPP7!/(4!2!)= 105 waysStage 2: Now lets place the 4I's-m-s-s-s-s-p-p-We can place th4 4I's in 8 waysTotal ways= 105*8= 840 ways. Is this a correct approach?
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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 *DIVIDED BY* (2)! for every letter repeated twice / (3)! for every letter repeated thrice and so on... Mississippi has: 4 s's4 i's2 p'sTotal number of letters: 11!So the answer is \(\frac{11!}{4!4!2!}\) _________________
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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 *DIVIDED BY* (2)! for every letter repeated twice / (3)! for every letter repeated thrice and so on... Mississippi has: 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.\(\frac{11!}{4!4!2!}\) gives us the total ways of arranging the 11 letters of the word MISSISSIPPI.Warm Regards,Pritish _________________
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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 B. 144 C. 210 D. 420 E. 840 Asked: In how many different ways can the letters of the word MISSISSIPPI be arranged if the vowels must always be together? 1. M-12. I - 43. S - 44. P - 2Vowels = I If 4 I s are to be together then IIII string should be considered a single letter1. M-12. IIII - 13. S - 44. P - 2No of arrangements = 8! / (4! * 2!) = 840 waysIMO E _________________
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Re: In how many different ways can the letters of the word MISSISSIPPI be [#permalink] 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 |
How many different ways can the letters of the following words be arranged a tarlac b Mississippi
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