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  • In how many ways can the letters of the word "MISSISSIPPI" be arranged such that all the vowels are not together?  <s

In how many ways can the letters of the word "MISSISSIPPI" be arranged such that all the vowels are not together?

 

Option: 1

12,589,620

 


Option: 2

39,899,280

 


Option: 3

10,859,638

 


Option: 4

20,475,820


Answers (1)

best_answer

To calculate the number of ways the letters of the word "MISSISSIPPI" can be arranged such that all the vowels (I, I, I, and I) are not together, we can calculate the total number of arrangements and subtract the number of arrangements where all the vowels are together.

Total number of arrangements of the letters in "MISSISSIPPI" is given by 11 !.

To calculate the number of arrangements where all the vowels are together, we can treat the group of vowels as a single entity. This means we have "MSSSSPPPIIII" to arrange.

Now, we have 8 entities to arrange, which are "M, S, S, S, P, P, I, I".

These 8 entities can be arranged in 8 ! ways. However, within this arrangement, the letter S repeats three times, the letter P repeats twice, and the letter I repeats four times. Therefore, we need to divide by 3 ! for the \mathrm{S^{\prime} s}, 2 ! for the P's, and 4 ! for the l's to account for the repetitions.

Therefore, the number of arrangements where all the vowels are together is:

\mathrm{8 ! /(3 ! \times 2 ! \times 4 !)=(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1) /(3 \times 2 \times 1 \times 2 \times 1 \times 4 \times 3 \times 2 \times 1)=5,040 / 288=17,520}

Finally, we can subtract the number of arrangements where all the vowels are together from the total number of arrangements to get the number of arrangements where all the vowels are not together:

\mathrm{ 11 !-17,520=39,916,800-17,520=39,899,280 . }

Therefore, there are 39,899,280 ways to arrange the letters of the word "MISSISSIPPI" such that all the vowels are not together.

Posted by

Divya Prakash Singh

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