This section covers permutations and combinations. Arranging Objects The number of ways of arranging n unlike objects in a line is n! (pronounced ‘n factorial’). n! = n × (n – 1) × (n – 2) ×…× 3 × 2 × 1 Example How many different ways can the letters P, Q, R, S be arranged? The answer is 4! = 24. This is because there are four spaces to be filled: _, _, _, _ The first space can be filled by any one of the four letters. The second space can be filled by any of the remaining 3 letters. The third space can be filled by any of the 2 remaining letters and the final space must be filled by the one remaining letter. The total number of possible arrangements is therefore 4 × 3 × 2 × 1 = 4!
n! . Example In how many ways can the letters in the word: STATISTICS be arranged? There are 3 S’s, 2 I’s and 3 T’s in this word, therefore, the number of ways of arranging the letters are: 10!=50 400 Rings and Roundabouts
When clockwise and anti-clockwise arrangements are the same, the number of ways is ½ (n – 1)! Example Ten people go to a party. How many different ways can they be seated? Anti-clockwise and clockwise arrangements are the same. Therefore, the total number of ways is ½ (10-1)! = 181 440 Combinations The number of ways of selecting r objects from n unlike objects is: Example There are 10 balls in a bag numbered from 1 to 10. Three balls are selected at random. How many different ways are there of selecting the three balls? 10C3 =10!=10 × 9 × 8= 120 Permutations A permutation is an ordered arrangement.
nPr = n! . Example In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10. Since the order is important, it is the permutation formula which we use. 10P3 =10! = 720 There are therefore 720 different ways of picking the top three goals. Probability The above facts can be used to help solve problems in probability. Example In the National Lottery, 6 numbers are chosen from 49. You win if the 6 balls you pick match the six balls selected by the machine. What is the probability of winning the National Lottery? The number of ways of choosing 6 numbers from 49 is 49C6 = 13 983 816 . Therefore the probability of winning the lottery is 1/13983816 = 0.000 000 071 5 (3sf), which is about a 1 in 14 million chance.
e-GMAT Representative Joined: 04 Jan 2015 Posts: 3758
How many 4-letter words can be formed using the alphabets of the word [#permalink] Updated on: 07 Aug 2018, 00:06
00:00
Difficulty: 45% (medium)
Question Stats: 65% (01:45) correct 35% (01:59) wrong based on 265 sessionsHide Show timer StatisticsHow many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?
E. \(240\) e-GMAT To read all our articles: Must read articles to reach Q51
Math Expert Joined: 02 Aug 2009 Posts: 10637
Re: How many 4-letter words can be formed using the alphabets of the word [#permalink] 30 Jan 2017, 08:33
EgmatQuantExpert wrote: How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?
E. \(240\) e-GMAT Hi ankujgupta Do not start arranging the alphabets before selection.Total 7 letters out of which 4 are to be choosen..G and L are already there, so choose 2 out of remaining 5.. 5C2=\(\frac{5!}{3!2!}=10\)..Now 4 letters can be choosen in 10 ways but they can be arranged in 4! Or 4*3*2=24 ways..Total 10*24=240..E _________________
e-GMAT Representative Joined: 04 Jan 2015 Posts: 3758
Re: How many 4-letter words can be formed using the alphabets of the word [#permalink] Updated on: 30 Jan 2017, 22:12
Manager Joined: 21 Jan 2016 Posts: 64 Location: India WE:Engineering (Computer Software)
Re: How many 4-letter words can be formed using the alphabets of the word [#permalink] 30 Jan 2017, 07:53 Is answer C ? We need to select 2 alphabets from E,N,I,S and H, which are arranged also, so 5P2 = 20. We have 3 places so total = 20*3*2*1 = 120;
Board of Directors Joined: 11 Jun 2011 Status:QA & VA Forum Moderator Posts: 5678 Location: India GPA: 3.5 WE:Business Development (Commercial Banking)
Re: How many 4-letter words can be formed using the alphabets of the word [#permalink] 30 Jan 2017, 08:15
EgmatQuantExpert wrote: How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?
E. \(240\) We have here 7 letters ( E,N,G,L,I,S & H )And 4 places as _ _ _ _G & L must be there, so we can arrange the 2 letters in 2! or 2 ways and the next 5 Letters ( E,N,I,S & H ) in 2 places in 5! ways ie, 120 ways...So, Total Number of ways is 120*2 = 240 ways..
Manager Joined: 21 Jan 2016 Posts: 64 Location: India WE:Engineering (Computer Software)
Re: How many 4-letter words can be formed using the alphabets of the word [#permalink] 30 Jan 2017, 10:27
chetan2u wrote: EgmatQuantExpert wrote: How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?
E. \(240\) e-GMAT Hi ankujgupta Do not start arranging the alphabets before selection.Total 7 letters out of which 4 are to be choosen..G and L are already there, so choose 2 out of remaining 5.. 5C2=\(\frac{5!}{3!2!}=10\)..Now 4 letters can be choosen in 10 ways but they can be arranged in 4! Or 4*3*2=24 ways..Total 10*24=240..E chetan2u Thanks. Got the error.
e-GMAT Representative Joined: 04 Jan 2015 Posts: 3758
How many 4-letter words can be formed using the alphabets of the word [#permalink] Updated on: 07 Aug 2018, 01:34
Hey, PFB the official solution. This question can be done in a number of ways. Let us focus on the two important methods of solving this question –Method 1 – Step 1: Understand the objective The objective of the question is to find the number of 4-letter words that can be formed from the alphabets of the word ENGLISH. The information given is:
• Repetition of letters is not allowed. • Two letters G and L are to be included necessarily in all the words. • Per the question, it is not necessary for the word formed to convey a meaning per English dictionary. Step 2: Write the objective equation enlisting all tasks The objective comprises of the following tasks:
Step 3: Determine the number of ways of doing each task
o Thus, number of ways to do Task 1 = 1
o Thus, number of ways to do Task 2 = 10
o Thus, number of ways to do Task 3 = 24 Step 4: Calculate the final answer
e-GMAT
e-GMAT Representative Joined: 04 Jan 2015 Posts: 3758
Re: How many 4-letter words can be formed using the alphabets of the word [#permalink] 30 Jan 2017, 22:12 Method 2 –
• We need to make 4 -letter words in which G,L must be present . • And the remaining two spaces must be filled with the remaining 5 letters. • Thus, the objective equation can be written as
• Thus, the remaining two spaces can be filled and arranged in = 5P2 = 5!/3! = 20 ways.• Plugging the two values in the objective equation, we get
• As we can see both the methods give us the same answer. In the first method, we first selected the letters and then arranged them and in the second method, we did the selection and arrangement simultaneously.
e-GMAT Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts _________________
Re: How many 4-letter words can be formed using the alphabets of the word [#permalink] 11 Dec 2018, 08:48
EgmatQuantExpert wrote: How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?
E. \(240\) Take the task of creating 4-letter words and break it into stages. Stage 1: Select 2 letters from E, N, I, S, H Since the order in which we select the two letters does not matter (yet!!), we can use combinations. We can select 2 letters 5 letters in 5C2 ways (10 ways)So, we can complete stage 1 in 10 ways ASIDE: If anyone is interested, we have a video on calculating combinations (like 5C2) in your head belowStage 2: Combine G and L with the two letters you chose in stage 1, and then arrange those 4 letters We can arrange n objects in n! waysSo, we can arrange the 4 letters in 4! ways (= 24 ways)We can complete stage 2 in 24 ways By the Fundamental Counting Principle (FCP), we can complete the two stages (and thus create 4-letter words) in (10)(24) ways (= 240 ways) Answer: ENote: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it. RELATED VIDEOS_________________
Brent Hanneson – Creator of gmatprepnow.comI’ve spent the last 20 years helping students overcome their difficulties with GMAT math, and the biggest thing I’ve learned is… Many students fail to maximize their quant score NOT because they lack the skills to solve certain questions but because they don’t understand what the GMAT is truly testing - Learn more
VP Joined: 04 Aug 2010 Posts: 1235 Schools:Dartmouth College
Re: How many 4-letter words can be formed using the alphabets of the word [#permalink] 26 Oct 2019, 13:32
EgmatQuantExpert wrote: How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?
E. \(240\)
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.Followed here and elsewhere by over 1900 test-takers.I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.As your tutor, I won't simply teach you how I would approach problems.I'll unlock the best way for YOU to solve problems.Available all over the world for virtual sessions and in NYC for live sessions. For more information, please email me at .
Senior Manager Joined: 10 Dec 2017 Posts: 253 Location: India
Re: How many 4-letter words can be formed using the alphabets of the word [#permalink] 26 Oct 2019, 14:20
EgmatQuantExpert wrote: How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?
E. \(240\) e-GMAT To read all our articles: Must read articles to reach Q51 E:)
Target Test Prep Representative Joined: 14 Oct 2015 Status:Founder & CEO Affiliations: Target Test Prep Posts: 16369 Location: United States (CA)
Re: How many 4-letter words can be formed using the alphabets of the word [#permalink] 01 Nov 2019, 18:37
EgmatQuantExpert wrote: How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?
E. \(240\) Since G and L must be used, the number of ways of choosing 2 more letters from the remaining 5 is 5C2 = (5 x 4)/2 = 10. However, once we have 4 letters, there are 4! = 24 ways to arrange them. Therefore, there are a total of 10 x 24 = 240 words that can be formed. Answer: E _________________
Senior Manager Joined: 26 May 2022 Posts: 286
Re: How many 4-letter words can be formed using the alphabets of the word [#permalink] 14 Jul 2022, 13:55 How many 4-letter words can be formed using the alphabets of the word…
9E30B0F1-E156-40AF-81DD-812627D7AA7C.jpeg [ 239.1 KiB | Viewed 5645 times ] _________________
Re: How many 4-letter words can be formed using the alphabets of the word [#permalink] 14 Jul 2022, 21:13 Asked: How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed? Since G & L are already selected, we have to select 2 letters out of remaining 5 and arrange them.Number of ways = 5C2 * 4! = 10 * 24 = 240 IMO E _________________
Kinshook Chaturvedi
Re: How many 4-letter words can be formed using the alphabets of the word [#permalink] 14 Jul 2022, 21:13 |