How many ways can the letter of the word promisebe arranged such that the vowels come together

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The letters of the word PROMISE are arranged so that no two of the vow [#permalink]

26 Nov 2018, 01:01

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The letters of the word PROMISE are arranged so that no two of the vowels should come together. Find total number of arrangements. A. 7B. 49 C. 1.440 D. 1,898

E. 4,320

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The letters of the word PROMISE are arranged so that no two of the vow [#permalink]

26 Nov 2018, 04:25

Really Good Question i learned it the hard way , hope my solution helps

__C__C__C__C__

Here is the arrangement of 4 Consonant such that between any two Consonant at most one Vowel can come but since the Vowels can also appear before the leftmost Consonant or after the rightmost Consonant so it gives us 5 places available for Vowels(represented by “___”)But since we require only 3 places due to 3 vowels to be arranged so we will select the 3 places out of 5 in 5C3 ways = 10 waysIn the given arrangement the Consonants can be arranged in 4! Ways at the selected 4 places and,All the vowels can also be arranged among themselves in 3! Waysso Total ways to arrange the letters as per desired condition = 10*3!*4! = 1440

Answer: Option C

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Re: The letters of the word PROMISE are arranged so that no two of the vow [#permalink]

26 Nov 2018, 12:41

This can also be solve as an overlapping set problem - 7!(total arrangements without restrictions) - 3.2.6!(arrangements when any 2 of three are together) + 5!.6 (removing one duplicate overlapping of all three together from previous as this will happen twice) = 1440.

Ans C.

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Re: The letters of the word PROMISE are arranged so that no two of the vow [#permalink]

26 Nov 2018, 17:18

Bunuel wrote:

The letters of the word PROMISE are arranged so that no two of the vowels should come together. Find total number of arrangements. A. 7B. 49 C. 1.440 D. 1,898

E. 4,320

Combine vowels together so we are left with 4 places for Consonants and 1 for vowels out of 7 we will have now 4+1 = 5 ways and since 3 vowels are given then the combination to arrange this array is 5c3

Now since no 2 vowels are to be together , so no. of ways to arrange C in 4 places 4 ! and vowels 3! = total arrangements hence possible = 5c3*4!*3! = 1440 option C

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Re: The letters of the word PROMISE are arranged so that no two of the vow [#permalink]

19 Aug 2019, 10:31

Let '__' denote the possible locations of the vowels in the resultant word.Since no two vowels can be together, there must be at least one consonant between them. __C1__C2__C3__C4__Number of ways in which consonants can rearrange among themselves = 4! = 24 waysNow, we have 5 different spots that can accommodate vowels.Number of vowels in PROMISE is 3.We have 5 spots; we need 3. Does order of appearance of vowels matter? Of course! Since we have to count possible arrangements here, order does matter.Therefore number of ways of choosing 3 spots from a pool of 5 (for our vowels) is 5P3 = 5!/2! = 60.Therefore total number of ways in which the acceptable arrangements can be achieved = 24*60 = 1440 ways _________________

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Wizard of Oddz
(Always do your best. What you plant now, you will harvest later.)

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Re: The letters of the word PROMISE are arranged so that no two of the vow [#permalink]

04 Oct 2019, 12:21

Bunuel wrote:

The letters of the word PROMISE are arranged so that no two of the vowels should come together. Find total number of arrangements. A. 7B. 49 C. 1.440 D. 1,898

E. 4,320

I have two solutions I wanted to share. 1. Using straight calculationWe know the set of consonants = C = {P, R, M, S} and set of vowels = V = {O, I, E}.Since no vowels should come together, all the consonants are placed in between them, or they form the start and end of the word. Hence the format for the acceptable answer is : _ C _ C _ C _ C _ Now, there are 4 consonants and we are using all the consonants. The way 4 consonants can be arranged when using all of them is 4!Talking about the vowels, we have 3 vowels and 5 place for them. So we have to pick 3 positions where we can put those 3 vowels. No. of ways we can pick 3 places from 5 available is 5C3.Once we have picked 3 places, we need to arrange three vowels in the 3 places picked. We can do that in 3! ways.Hence total no. of ways is : 4 ! * 5C3 * 3 ! = 14402. Using negation

no. of ways so that no two of the vowels are together = total no of ways - no. of ways so all 3 vowels are together - no. of ways so 2 of the vowels are together

total no of ways we can arrange C + V = 7 ! total no. of ways we can find when 3 vowels are together : Consider 3 vowels to be 1 new letter σ. Now we have 4 letters of consonants and σ making it 5 letters. We can arrange 5 letters in 5 ! ways. But σ has 3 letters inside and each arrangement between those 3 letters gives 1 new solution. So total solution : 5! * 3! total no. of ways we can find when 2 vowels are together : We choose 2 vowels to form a new letter ζ. That can be done in 3C2 ways. Now we can choose 1 place from the 5 possible place for ζ by 5C1 . Within ζ, there can be 2! arrangementsThen there is 1 vowel remaining and 4 places remaining which we can arrange in 4 different ways.So 7! - 5!*3! - (3C2 * 5C1 * 2! * 4) = 1440

I know 2nd option becomes cumbersome but was a great exercise in thinking through different way to do the problem.

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Re: The letters of the word PROMISE are arranged so that no two of the vow [#permalink]

13 Jun 2021, 09:18

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Re: The letters of the word PROMISE are arranged so that no two of the vow [#permalink]

13 Jun 2021, 09:18