How many ways can the letters PQRS and T are arranged so that P and T are always together?
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Math Expert Joined: 02 Sep 2009 Posts: 86947 The letters D, G, I, I , and T
can be used to form 5-letter strings as [#permalink]
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Hide Show timer StatisticsThe letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter? A) 12 _________________ Originally posted by Bunuel on 15 Jun 2016, 00:51. Cleaned topic. Math Expert Joined: 02 Sep 2009 Posts: 86947 Re: The letters D, G, I, I , and T can be used to form 5-letter strings as
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skysailor wrote: Could someone help me understand how the total number of cases is 5!/2!? I don't understand the reasoning behind that. THEORY: Permutations of \(n\) things of which \(P_1\) are alike of one kind, \(P_2\) are alike of second kind, \(P_3\) are alike of third kind ... \(P_r\) are alike of \(r_{th}\) kind such that: \(P_1+P_2+P_3+..+P_r=n\) is: \(\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}\). For example number of permutation of the letters of the word "gmatclub" is 8! as there are 8 DISTINCT letters in this word. Number of permutation of the letters of the word "google" is \(\frac{6!}{2!2!}\), as there are 6 letters out of which "g" and "o" are represented twice. Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be \(\frac{9!}{4!3!2!}\). For more check the links below: Combinatorics Made Easy! Theory on Combinations DS questions on Combinations Tough and tricky questions on Combinations So, as
explained above, the number of arrangements of 5 letters D, G, I, I , and T out of which there are two identical I's will be 5!/2! = 60. GMAT Club Legend Joined: 11 Sep 2015 Posts: 6829 Location: Canada
Re: The letters D, G, I, I , and T can be used to form 5-letter strings as
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Bunuel wrote: The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter? A) 12 Another approach. Take the task of arranging the 5 letters and break it into stages. Stage 1: Arrange the 3 CONSONANTS (D, G and T) in a row IMPORTANT: For each arrangement of 3 consonants, there are 4 places where the two I's can be placed. Stage 2: Select two available spaces and place an I in each space. By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus arrange all 5 letters) in (6)(6) ways (= 36 ways) Answer: D Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So be sure to learn this technique. RELATED VIDEOS _________________ Brent Hanneson – Creator of gmatprepnow.com Manager Joined: 04 Apr 2015 Posts: 91 Re: The letters D, G, I, I , and T can be used to form 5-letter strings as [#permalink]
IMO 36 Total no of ways arranging 5 letter with one letter redundant is 5!/2!=60 no of ways at least one alpha is between two I's =60-24=36 Manager Joined: 18 Jan 2010 Posts: 229 Re: The
letters D, G, I, I , and T can be used to form 5-letter strings as [#permalink]
Bunuel wrote: The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter? A) 12 Let us calculate the total ways. Those would be (5!/2!) = 60. Now since the question says "at least" let us find the number of arrangements when both I's are together. (Tie them up). so we have 4! ways to arrange such that I's always come together. 4! = 24 60 - 24 = 36. D is the answer. Current Student Joined: 18 Oct 2014 Posts: 724 Location: United States GPA: 3.98
Re: The letters D, G, I, I , and T can be used to form 5-letter strings as [#permalink]
Bunuel wrote: The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter? A) 12 Total ways to arrange these letters=5!/2! Ways to arrange the letters if both I are together= 4! Ways to arrange with both Is not together= 5!/2!-4!= 36 D is the answer I welcome critical analysis of my post!! That will help me reach 700+ SVP Joined: 06 Nov 2014 Posts: 1816 Re: The letters D, G, I, I , and T can be used to form 5-letter strings as
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Bunuel wrote: The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter? A) 12 Required: atleast one letter between two Is Total cases = 5!/2! = 60 Hence total cases in which there is atleast one letter between the Is = 60 - 24 = 36 Correct Option: D Manager Joined: 30 Jul 2013 Posts: 60 Concentration: Technology, General Management GMAT Date: 07-03-2015 GPA: 3.8 WE:Information Technology (Computer Software)
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The requirement is 5 letter strings in which two I's are not together, they are separated by atleast one letter. Consider this as below, (Total number of 5 letter strings that can be formed) - (Total number of 5 letter strings in which two I's are together) Finding total number of 5 letter strings - DIGIT = 5!/2! (2! because of the repetition of I's) Manager Joined: 25 Jun 2016 Posts: 59 Re: The letters D, G, I, I , and T can be used to form 5-letter strings as [#permalink]
A couple of video explanations for this question: "Direct" Method: "Indirect" Method: Intern Joined: 09 Apr 2017 Posts: 26 Location: United States GMAT 1: 690 Q47 V38 GMAT 2: 720 Q48 V41 GPA: 3.5 Re: The letters D, G, I, I , and T can be used to form 5-letter strings as [#permalink]
Could someone help me understand how the total number of cases is 5!/2!? I don't understand the reasoning behind that. Director Joined: 17 Dec 2012 Posts: 611 Location: India Re: The letters D, G, I, I , and T can be used to form 5-letter strings as
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Bunuel wrote: The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter? A) 12 1. It is an ordering, therefore a permutation problem Intern Joined: 12 Jan 2017 Posts: 33 Location: India Concentration: Operations, Leadership Re: The letters D, G, I, I , and T can be used to form 5-letter strings as
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The approach i used was...I combined both I's together,so now there are four letters which can be arranged in 4! ways. Total number of ways 5 letters can be arranged without any constraints is 5!. So the the answer should be 5!-4! as ...can someone help to explain what wrong i did? Math Expert Joined: 02 Aug 2009 Posts: 10621
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sandaki wrote: The approach i used was...I combined both I's together,so now there are four letters which can be arranged in 4! ways. Total number of ways 5 letters can be arranged without any constraints is 5!. So the the answer should be 5!-4! as ...can someone help to explain what wrong i did? Hi You are wrong in coloured portion.. Target Test Prep Representative Joined: 14 Oct 2015 Status:Founder & CEO Affiliations: Target Test Prep Posts: 16330 Location: United States (CA) Re: The letters D, G, I, I , and T can be used to form 5-letter strings as [#permalink]
Bunuel wrote: The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter? A) 12 This is a permutation problem because the order of the letters matters. Let’s first determine in how many ways we can arrange the letters. Since there are 2 repeating Is, we can arrange the letters in 5!/2! = 120/2 = 60 ways. We also have the following equation: 60 = (number of ways to arrange the letters with the Is together) + (number of ways without the Is together). Let’s determine the number of ways to arrange the letters with the Is together. We have: [I-I] [D] [G] [T] We see that with the Is together, we have 4! = 24 ways to arrange the letters. Thus, the number of ways to arrange the letters without the Is together (i.e., with the Is separated) is 60 - 24 = 36. Answer: D See why Target Test Prep is the top rated GMAT course on GMAT Club. Read Our Reviews EMPOWERgmat Instructor Joined: 19 Dec 2014 Status:GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Posts: 21005 Location: United States (CA) Re: The letters D, G, I, I , and T can be used to form 5-letter strings as [#permalink]
Hi All, Since the two "I"s cannot be side-by-side, there are a limited number of ways to arrange the 5 letters. As such, with a little permutation math and some 'brute force', we can map out the possibilities: _ _ _ _ _ If the first letter is an I, then the second letter CANNOT be an I (it would have to be one of the other 3 non-I letters)... i 3 From here, any of the remaining letters can be in the 3rd spot. After placing one, either of the remaining two letters can be in the 4th spot and the last letter would be in the 5th spot... i 3 3 2 1 This would give us (3)(3)(2)(1) = 18 possible arrangements with an I in the 1st spot. If a non-I is in the 1st spot and an I is in the 2nd spot, then we have... 3 i _ _ _ A non-I would have to be in the 3rd spot, then either remaining letter could be 4th... 3 i 2 2 1 This would give us (3)(2)(2)(1) = 12 possible arrangements Next, we could have two non-Is to start off, then Is in the 3rd and 5th spots... 3 2 i 1 i This would give us (3)(2)(1) = 6 possible arrangements There are no other options to account for, so we have 18+12+6 total arrangements. Final Answer: GMAT assassins aren't born, they're made, Intern Joined: 18 Apr 2013 Posts: 32 Re: The letters D, G, I, I , and T can be
used to form 5-letter strings as [#permalink]
Hi Bunuel, can I check why the no of ways when the Is are placed together is 4!? It can be located _D_G_T_. Why isn't it 4 ways? Thanks in advance! Math Expert Joined: 02 Sep 2009 Posts: 86947 Re: The letters D, G, I, I , and T can be used to form 5-letter strings as
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roastedchips wrote: Hi Bunuel, can I check why the no of ways when the Is are placed together is 4!? It can be located _D_G_T_. Why isn't it 4 ways? Thanks in advance! In addition to that D, G, and T, could be arranged in 3! ways, so total = 4*3! = 4!. Or, consider two I's as one unit {II}. We'll have 4 units: {D}, {G}, {T}, and {II}. The number of arrangements is 4!. Hope it's clear.
GMAT Club Legend Joined: 11 Sep 2015 Posts: 6829 Location: Canada
Re: The letters D, G, I, I , and T can be used to form 5-letter strings as [#permalink]
Bunuel wrote: The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter? A) 12 When answering counting question, one should always consider whether a useful approach would be listing and counting the possible outcomes. How do we know when this approach may be the best approach? TLDR: Don't dismiss listing and counting as a possible approach to answering counting questions. POSSIBLE OUTCOMES Done! Did that take under 2 minutes? Probably. Cheers, Brent Hanneson – Creator of gmatprepnow.com VP Joined: 10 Jul 2015 Status:Expert GMAT, GRE, and LSAT Tutor / Coach Affiliations: Harvard University, A.B. with honors in Government, 2002 Posts: 1045 Location: United States (CO) Age: 42 GMAT 1: 770 Q47 V48 GMAT 2: 730 Q44 V47 GMAT 3: 750 Q50 V42 GMAT 4: 730 Q48 V42 (Online) GRE 1: Q168 V169 GRE 2: Q170 V170 WE:Education (Education) Re: The letters D, G, I, I , and T can be used to
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The correct answer is Choice D. First, determine the total number of ways of rearranging the letters in DIGIT. 5! / 2! (accounts for the repetition of the 2 letter "I"s) = (5 x 4 x 3 x 2) / 2 = 120 / 2 = 60 Next, consider all the ways two letter "I"s can be adjacent within the word. They can either be in the 1/2 slot, the 2/3 slot, the 3/4 slot, or the 4/5 slot (4 locations), and for each of those 4 locations there are 3 x 2 x 1 = 6 other ways of rearranging the final 3 letters, so multiply 6 by 4 to get 24. Subtract those 24 instances from the total to get 60 - 24 = 36 -Brian My name is Brian McElroy, founder of McElroy Tutoring (https://www.mcelroytutoring.com). I'm a 42 year-old Providence, RI native, and I live with my wife, our three daughters, and our two dogs in beautiful Colorado Springs, Colorado. Since graduating from Harvard with honors in the spring of 2002, I’ve worked full-time as a private test-prep tutor, essay editor, author, and admissions consultant. I’ve taken the real GMAT 6 times — including the GMAT online — and have scored in the 700s each time, with personal bests of 770/800 composite, Quant 50/51, Verbal 48/51, IR 8 (2 times), and AWA 6 (4 times), with 3 consecutive 99% scores on Verbal. More importantly, however, I’ve coached hundreds of aspiring MBA students to significantly better GMAT scores over the last two decades, including scores as high as 720 (94%), 740 (97%), 760 (99%), 770, 780, and even the elusive perfect 800, with an average score improvement of over 120 points. I've also scored a verified perfect 340 on the GRE, and 179 (99%) on the LSAT. Intern Joined: 12 Nov 2019 Posts: 4 Re: The letters D, G, I, I , and T can be used to form 5-letter strings as
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varundixitmro2512 wrote: IMO 36 Total no of ways arranging 5 letter with one letter redundant is 5!/2!=60 no of ways at least one alpha is between two I's =60-24=36 can you please explain why the No of ways two I's can be together is 4!=24? Re: The letters D, G, I, I , and T can be used to form 5-letter strings as [#permalink] 20 Nov 2019, 15:58 Moderators: Senior Moderator - Masters Forum 3088 posts How many ways can be the letters PQRS and T arranged so that P and T are always together?These can be arranged in 6! ways.
How many different ways can the letters P Q R S be arranged?How many different ways can the letters P, Q, R, S be arranged? The answer is 4! = 24. The first space can be filled by any one of the four letters.
How many ways can the letters in the word parallel be arranged if the P and R are together?Solution : The given word 'PARALLEL' has 8 letters, out of which there are 2 A's, 3 L's, 1 P, 1 R and 1 E.
Number of their arrangements `=(8!)/((2!) xx(3!)) =3360. How many ways can you arrange the letters A B C find all the ways?You can arrange the letters A, B, and C in different ways: ABC, ACB, and so on. An arrangement in which order is important is a permutation. There are 6 possible arrangements.
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