Exercise :: Permutation and Combination - General Questions
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| 13. | In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together? | |||||||||||||||
Answer: Option C Explanation:
In the word 'MATHEMATICS', we treat the vowels AEAI as one letter. Thus, we have MTHMTCS (AEAI). Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.
Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.
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Exercise :: Permutation and Combination - General Questions
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| 7. | How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated? |
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Answer: Option D Explanation:
Since each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it. The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place. The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it. |
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How many 5 letter words ( with or without meaning) can be formed using [#permalink]
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How many 5 letter words ( with or without meaning) can be formed using all the following 5 letters P,Q,R,S,and T so that letter P is to the left of letter R?(A) 120(B) 60(C) 48(D) 24
(E) 12
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Re: How many 5 letter words ( with or without meaning) can be formed using [#permalink]
It seems there are only 4 distinct letters in the question stem .
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Re: How many 5 letter words ( with or without meaning) can be formed using [#permalink]
AryamaDuttaSaikia wrote:
How many 5 letter words ( with or without meaning) can be formed using all the following 5 letters P,Q,R,S,and T so that letter P is to the left of letter R?(A) 120(B) 60(C) 48(D) 24
(E) 12
We can solve this qustion by 2 methods,Method 1: Total combinations for 5 letters = 5! = 120
As there is no bias in counting the combinations, half of the combinations will have R to the left of P and half of them to the right of P.Thus possible combinations = 120/2 = 60. B is the correct answer.Method 2:
The combinations possible are:PRQST, combinations of RQST = 4! =24QPRST, combinations = 3C1*3!, 3C1 taken to account for the fact that instead of Q we can also take S or TQSPRT, combinations = 2!*2!*3C2, 3C2 taken to account for the fact that instead of QS we can also take ST or QT, 2! each to take into account permutations for QS and RTQSTPR, combinations = 3!*1, 3! to account for permutations for QSTThus, the total combinations possible = 4!+3C1*3!+2!*2!*3C2+3! = 24+18+12+6=60. B is the correct answer. _________________
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Re: How many 5 letter words ( with or without meaning) can be formed using [#permalink]
AryamaDuttaSaikia wrote:
How many 5 letter words ( with or without meaning) can be formed using all the following 5 letters P,Q,R,S,and T so that letter P is to the left of letter R?(A) 120(B) 60(C) 48(D) 24
(E) 12
Best Method:
Total Ways of arranging 5 letters in any possible order = 5*4*3*2*1 = 5! = 120in Half of the cases P will be to the left of R and in other half P will be to the right of RHence, Desired outcome = 120/2 = 60Alternate method:
we have five blank spaces _ _ _ _ _ which have to occupied by these 5 letter such that P is to the left of Rso let's select two blank spaces for P and R out of 5 blank spaces, No. of ways of selecting two blank spaces out of 5 = 5C2 = 10The no. of ways of arranging P and R on selected two places such that P is to the left of R = 1No. of ways of arranging remaining three letters on remaining three blank spaces = 3*2*1 = 3! = 6Total Ways of Arrangement = 5C2 * 1 * 3! = 10 * 1 * 6 = 60Answer: option B _________________
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Re: How many 5 letter words ( with or without meaning) can be formed using [#permalink]
Conceptual wayAs a general strategy in basic Question on Permutation Combination like this we first arrange the letters on which there is a condition and then arrange the remaining letters. So out of the five positions where the 5 letters needs to be arranged we will first arrange letters P and R. So letters P and R will take two positions so lets select two positions out of 5 positions where letters P and R will be placed Two positions can be selected out of 5 positions in 5C2 ways = 10 ways. Once two positions have been selected we know P will be in the left position and R will be in right so there is only one way we can place letters P and R in these two positions. And the remaining 3 letters can be placed in 3 posions in 3! Ways = 6 ways. So the final Answer = number of ways letters P and R can be arranged x number of ways the other 3 letters can be arranged Final Answer = 10 x 6 = 60 ways
JAMBOREE Way
Now 5 letters can be placed in 5 positions in 5! Ways = 120. Logically we can analyze that either letter P will be to the left of letter R or to the right of letter R. So by simple reasoning we will get that in half of the ways letter P will be to the left of letter R.So the correct answer = ½ total number of ways = 60 waysWe make the GMAT simple!
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How many 5 letter words ( with or without meaning) can be formed using [#permalink]
Why can't I consider PR as one letter? So I only have 4 letters which can be rearranged.(PR) Q S T n = 4! = 24?
Where is my mistake? I don't understand it
Thank you very much.
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Re: How many 5 letter words ( with or without meaning) can be formed using [#permalink]
22gmat wrote:
Why can't I consider PR as one letter? So I only have 4 letters which can be rearranged.(PR) Q S T n = 4! = 24?
Where is my mistake? I don't understand it
Thank you very much.You have only considered the case in which R is to the immediate right of P , but as per the question stem R can take any of the positions to right of P ( other letters can come in between)P _ _ _ _R can take any of the 4 positions as long as it satisfies being to right of P.PRQST , Combination of RQST =4!=24whereas as per your procedure , (PR) Q S T , there will be 3! combinations for this configuration.And you can proceed similarly for the other configurations .QPRST-3C1*3!QSPRT- 2!*2!*3C2QSTPR- 3!*1
Hope it helped !!
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Re: How many 5 letter words ( with or without meaning) can be formed using [#permalink]
22gmat wrote:
Why can't I consider PR as one letter? So I only have 4 letters which can be rearranged.(PR) Q S T n = 4! = 24?
Where is my mistake? I don't understand it
Thank you very much.
You might find this post useful: //www.veritasprep.com/blog/2011/10 ... s-part-ii/
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Re: How many 5 letter words ( with or without meaning) can be formed using [#permalink]
Thank you all!
Got it
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Re: How many 5 letter words ( with or without meaning) can be formed using [#permalink]
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Re: How many 5 letter words ( with or without meaning) can be formed using [#permalink]
28 Jun 2021, 02:42