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In how many ways we can arrange the letters of the the word MANIFEST with vowels in alphabetical order?

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Only 6! can be the answer and nothing else. The logic is :consider AEI as 1 separate unit ( as one letter). Apart from it, 5 letters are remaining. So, in total there are 6 letters that have to be arranged. These can be arranged in 6! ways.
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No need, but required practice in any language and practical knowledge if you go for programming side.
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IIT + IIM Tutor

Group AEI into one group and treat them as one alphabet. Now you have M, N, F, S, T, (AEI) - 6 alphabets. You can arrange 6 alphabets in 6! ways. So your answer is 6x5x4x3x2x1 = 720 ways
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6!
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Competitive Exam Preparation

6!.. Consider AEI as one unit. Then remains 5. Now you need to arrange 6 so ans would be 6!
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It is 8!/3! 8! ways all letters are arranged which includes 3! ways AIE can be arranged within themselves. Since they should be in alphabetical order divide by 3!
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Mathematics ( Quantitative Aptitude) Trainer

6! ways. AEI can be grouped as 1 and the other 5 words So this can be arranged in 6! ways
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An Adept Mentor & Coach. Expert in Marketing/Branding, Math and Quantitative Aptitude. Post Graduate in Management from one of the Top-5 MBA College in India.

It should be 8!/3!. All 8 letters can be arranged in 8! ways... however AEI can be arranged in only one way.. hence we can treat them as identical letters. So divided by 3!
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calculate using permutation 8!= 8*7*6*5*4*3*2*1
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