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  • In how many ways can the letters of the word "MISSISSIPPI" be arranged such that all the vowels are together?  <stron

In how many ways can the letters of the word "MISSISSIPPI" be arranged such that all the vowels are together?

 

Option: 1

125


Option: 2

150


Option: 3

100


Option: 4

200


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 together, we can treat the group of vowels as a single entity. This means we have "MSSSSPPP(IIII)" to arrange.

Now, we have 7 entities to arrange, which are " M, S, S, S, S, P, P ".

These 7 entities can be arranged in \mathrm{7 !} ways. However, within this arrangement, the letter S repeats

four times and the letter P repeats twice. Therefore, we need to divide by \mathrm{4 !} for the S's and \mathrm{2 !} for the P's to account for the repetitions.

Additionally, within the group (IIII), the vowels I can be arranged among themselves in \mathrm{ 4 !} ways.

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

\mathrm{ (7 ! /(4 ! \times 2 !)) \times 4 !=(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1) /(4 \times 3 \times 2 \times 1 \times 2 \times 1) \times 24=5,040 / 48 \times 24=5,040.}

Thus, there are 5,040 ways to arrange the letters of the word "MISSISSIPPI" such that all the vowels are together.

Posted by

manish painkra

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