Aptitude - Permutation and Combination - Discussion

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"Two things are infinite: the universe and human stupidity; and I'm not sure about the universe."
- Albert Einstein

In how many different ways can the letters of the word 'LEADING' be arranged in such a way that the vowels always come together?

[A]. 360[B]. 480
[C]. 720[D]. 5040
[E]. None of these

Answer: Option


The word 'LEADING' has 7 different letters.

When the vowels EAI are always together, they can be supposed to form one letter.

Then, we have to arrange the letters LNDG (EAI).

Now, 5 (4 + 1 = 5) letters can be arranged in 5! = 120 ways.

The vowels (EAI) can be arranged among themselves in 3! = 6 ways.

Required number of ways = (120 x 6) = 720.

Shoba said: (Sat, Sep 4, 2010 06:31:36 AM)    
Other than vowels there are only 4 letter then how it s possible to get 5!.

Sai said: (Fri, Sep 10, 2010 02:26:28 AM)    

4 consonants + set of vowels (i. E. , L+N+D+G+ (EAI) ).

We should arrange all these 5. So we get 5!.

I think you understood.

Sachin Kumar said: (Wed, Sep 29, 2010 08:40:31 AM)    
Please make me understand this answer as did not get.

Sri said: (Sun, Oct 10, 2010 09:12:07 AM)    
As vowels are together take (EAI) as single letter i.e. , total no of letters are 5 (L, N, D, G, {EAI}).

No of ways can arrange these 5 letters are 5! ways.

Now we arranged 5 letters (L, N, D, G, {EAI}).

Next we have to arrange E, A, I (they may be EAI/EIA/AEI/AIE/IAE/IEA).

All these combinations imply that vowels are together.

So we have to multiply 5! and 3!.

Subbu said: (Thu, Jan 6, 2011 03:09:54 AM)    

Madhusudan said: (Mon, Mar 14, 2011 10:00:52 AM)    
Could you kindly let me know what is ( ! ).Howe 5! = 120 ?

Sundar said: (Mon, Mar 14, 2011 10:38:38 AM)    

5! = 5 Factorial = 1 x 2 x 3 x 4 x 5 = 120

Laxmikanth said: (Sat, Mar 26, 2011 11:54:29 PM)    
When should we take the one or more letters as a single unit and why?

Moaned said: (Fri, Apr 8, 2011 10:34:06 AM)    
I am not getting

Jessie said: (Sun, Apr 10, 2011 05:47:38 AM)    
7 letter word = LEADING
permuation to arrange 5 letters = nPr= n!/(n-r)!=5!/0!=5!
0! is assumed to be 1!)
EAI can be arranged among each other in = nPr = 3!/(3-3)= 3!
hence 5! x 3! = 120 x 6 = 720

Mayur said: (Fri, Apr 15, 2011 02:30:58 AM)    
How it came like nPr formula and how you have solve it? please let me know.

Bharti said: (Mon, Jul 11, 2011 06:54:15 AM)    
Thanks a lot. I had confusion before your explanations. Thanks a lot.

Pavan@9966606261 said: (Fri, Aug 5, 2011 12:14:00 PM)    

You just look when ever you open the new exercise there will be availability of basic formulas. If you go through them half of the task would be finished easily. Have a good day buddy.

Siva said: (Sun, Oct 16, 2011 12:36:43 PM)    
Permutations and combinations always make me to confuse much.

How to decide based on the descriptive aptitude question ?

Shiva said: (Tue, Oct 18, 2011 08:30:16 PM)    
Why we have taken 5 (4 + 1) ?

Santosh Kumar Pradhan said: (Thu, Oct 27, 2011 01:00:33 PM)    
Total member 7
out of which 5 consonant and 3 vowels
take 3 vowls as a 1
hen number of consonant will arrange 5! ways
and these 3 vowels will arrange 3! ways

Bhavik said: (Fri, Nov 4, 2011 11:28:22 AM)    
How to read these nPr ?

Sagar Choudhary said: (Sun, Nov 6, 2011 07:02:12 PM)    
What is this 5! and 3! ?

A.Vamsi Krishna said: (Mon, Jan 30, 2012 06:18:08 AM)    
"!" this implies factorial that means a number is multiplied
like for example take number 5 then its factorial will be
taken as 5*4*3*2*1 and this is equal to 120.

Ncs said: (Thu, Feb 2, 2012 05:55:31 PM)    
Why not 5! + 3! ?

Raj said: (Mon, May 21, 2012 01:35:02 PM)    
Friends, How do you say this question is permutation.

Mahanthesh said: (Mon, Jul 30, 2012 11:54:44 PM)    
Hi guys. As per the question, it s mentioned that all vowels should be together, but it has not been mentioned that it should be EAI. According to me these 3 vowels can be arranged in 3*2=6 ways. And the remaining letters LDNG can be arranged in 4*3*2*1= 24 ways. Can anyone explain me about this please ?

Sandeep said: (Fri, Sep 7, 2012 10:49:23 AM)    
can any one explain ? y 5!*3! & y not like this 5!+3!

Srk said: (Tue, Oct 9, 2012 09:52:35 AM)    

1. There is given a word LEADING in this LDNG (consonents) , EAI (vowels).

2. They asked here vowels always come together and so we should have to take LDNG (EAI).

3. We can take as (4+1) ! i.e. here we have to take vowels as together as 1. So we have a chance of 5!.

4. But with in vowels we have many arrangements i.e 3!

5. Finally 5!*3!

Shafi said: (Thu, Oct 11, 2012 11:02:09 PM)    

You are correct 3! and 4!, because you spitted vowels and consonants,

But to combine them vowels+consonants.

i.e. (4 consonants + 1 set of vowels) = 5!

And (3 vowels {1 set}) = 3!

So 5!*3! = 5x4x3x2x1x3x2x1=720.

Adhityasena said: (Sat, Feb 9, 2013 09:47:03 AM)    
I get the answer to be 2*720. Because, since the vowels must come together in the word LEADING, which actually has 7 letters, E and A can be taken as one unit, so now we have L, EA, D, I, N, G to be arranged and which can be done in 6! ways. But among E and A there are 2 arrangements namely EA and AE, So the final answer is 2*720. Am I right !

Chetas said: (Sat, Jul 6, 2013 03:05:23 PM)    

You have not considered a vowel 'I'. 'EAI' is to be taken as one unit.

Sanjay(9776387850) said: (Thu, Aug 29, 2013 08:31:19 PM)    
Hi Guys, at first I tell you how can you know is this permutation or combination. Permutation means arrange(row/column) and Combination means selection(group).

i.e. Number and Word are perm. Playing 11 and committee are combi. This is a word so this is perm. You should know the formula that m different objects are alike and n different object are alike if we arrange all the m+n objects such that n objects are always together=(m+1)!*n!

Here n = Vowel = 3 and m = Con. = 4.

So = (4+1)!*3! = 5!*3!.

= 120*6 = 720.

Chithra said: (Tue, Nov 5, 2013 09:08:27 PM)    
As we known very well that * is consider to be AND, + is consider to be OR. Consider the 1st example 7 men and 6 women problem we taken as (7c3*6c2) + (7c4*6c2) + (7c5).

We can also said this as (7c3 AND 6c2) OR (7c4 AND 6c2) OR (7c5).

CONDITION -> 5 people need to select. We should take 3 men from 7 men and 2 women from 6 women or other option is to take 4 men from 7 men and 1 women from 6 women.

That's why we are using * at the place of AND, + at the place of OR. Similarly, we need to consider both the consonant AND vowel not consonant OR vowel.

So we use 5!*3!.

Taku Mambo Bellnuisemarbel said: (Fri, Nov 29, 2013 03:04:05 PM)    
Please help me with this; whenever I hear of probability that a variable is being selected what should I think of?

Amna Fida said: (Tue, Dec 3, 2013 10:46:51 PM)    
How we know that factorial will use here?

Amnafida said: (Tue, Dec 3, 2013 10:50:35 PM)    
Describe that how we know about here permutation is use?

Jhansi Sri said: (Fri, Feb 7, 2014 10:48:37 AM)    
Please help me quickly why we take 5!*3!, Why we can't take 5!+3!.

Pema said: (Sun, Aug 3, 2014 09:03:16 AM)    
When it comes the question for arrangements, then it is a Permutation Or you all can remember it as keyword "PA" P=permutation and A=arrangement.

Likewise, for combination, it is all for selection purpose, remember keyword as "CS" c=combination,s=selection. Then apply formula for each. Easy.

Baidyanath Jena said: (Fri, Nov 7, 2014 08:53:38 PM)    
When it comes to persons it should be combination.

Samson said: (Mon, Nov 17, 2014 09:52:04 AM)    
God bless you all for your contribution especially you @Jessie for using the formula to break it down well.

Ranjeet said: (Wed, Nov 19, 2014 06:43:25 PM)    
Well I am confused. Somewhere n! is done whereas somewhere (n-1)! is used.

Can someone explain about it?

Sagar said: (Fri, Dec 26, 2014 03:33:53 PM)    
Hi friends.

We know that formula n!=n (n-1) (n-2).....3.2.1. Suppose there n way to choose first element (since there are n elements).

After that there are n-1 ways to choose second element because already we choose one element from n elements that's why we are assuming this way. Similarly n-2 ways to chose the third element..etc it's going like this.

n!=n (n-1).

n!=n (n-1) (n-2) if n>2 or equals 2.

n!=n (n-1) (n-2) (n-3) if n>3 or equals 3.

Hope you understood.

Shantha said: (Thu, Apr 30, 2015 11:00:35 PM)    
Then how we won't take E+A+I+(LDNG) = 4.

Tom said: (Sat, May 2, 2015 11:10:52 PM)    
In what situations we can permutation or combination?

Riya said: (Fri, May 8, 2015 11:26:25 AM)    
@Tom and @Shantha.

Whenever there is a reference to some arrangement it is permutation. Whenever there is a reference to some selection it is combination.

The given question requires us to find the number of ways in which the word LEADING can be arranged with the condition that the vowels (EAI) always be together. Thus we need to apply the concept of permutation.

Here, since the vowels EAI must always be together we consider it as a single word (EAI).

Thus, LDNG (EAI) make a 5 letter word.

It can be arranged in 5p5 ways = 5!ways.

Now, since EAI can arrange itself in 3p3 or 3! ways, the word LDNG (EAI).

Can thus be arranged or PERMUTED in 3!*5! ways = 720 ways.

Spurthy said: (Wed, Jun 24, 2015 10:15:47 PM)    
All the consonants can also be written as a unit?

Ajay said: (Mon, Aug 17, 2015 09:13:01 PM)    
In some cases unit's place, tens place's and hundred's place are used what its mean?

Shrinivas said: (Tue, Aug 25, 2015 08:11:12 PM)    
Hi if two vowels are repeated in same word then will you take it as same or as different?

Vanmathi said: (Tue, Sep 1, 2015 10:35:56 PM)    
In how many ways LEADING be arranged such a way that atleast two vowels always together?

Soumya Sengupta said: (Fri, Sep 18, 2015 12:01:46 PM)    
If there is another vowel 'o' what should be done like in 'outstanding' here a, i, o, u are vowels do we have to consider (aiou) = 1 letter for calculation?

Palak said: (Fri, Sep 18, 2015 06:33:52 PM)    
In this question won't the arrangement of non vowels matter and why?

Raja said: (Sat, Sep 19, 2015 02:19:57 PM)    
Hai @Soumya.

S its 1 letter only. Hope you understand.

Yonatan said: (Thu, Sep 24, 2015 02:10:52 PM)    
I can't understand the answer.

Mukesh said: (Fri, Dec 18, 2015 04:18:24 PM)    
Please anyone help me why repetition is not considered in the current problem?

Eswaru said: (Wed, Feb 3, 2016 01:39:56 PM)    
Hey dude we have to form 7 letter words from LEADING that means we need to use all the letters at a time. So no repetition are allowed.

Peter said: (Sun, Feb 28, 2016 01:31:15 PM)    
If at least two vowels always together then what will be the answer for LEADING ?

Siva said: (Wed, Mar 16, 2016 02:32:11 PM)    
Other than vowels there are only 4 letter then how it s possible to get 5!.

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