Number of words in which all the vowels are not together of the word GANESHPURI?

The options available are

  1. 21×7!
  2. 42×8!
  3. 84×7!
  4. None

*I have found words with vowels always together=7!×4! and subtracted total number of words from it(10!-7!×4!) but it seems wrong. *

Also; 2:) Number of words with any two of the letters E,H and P are never together?

  • $\begingroup$ To clarify, do you mean "it is not the case that all vowels are together" where AEUGNSHPRI counts as a valid arrangement since the I is not a part of the block with the rest of the vowels, or do you mean "it is the case that no vowel is next to any other vowel" where it would not have been a valid string since you have the A and the E adjacent among others. $\endgroup$ – JMoravitz Mar 13 at 16:57
  • $\begingroup$ @JMoravitz Given the phrasing of the second question, the first interpretation seems sound. $\endgroup$ – N. F. Taussig Mar 13 at 16:58
  • 1
    $\begingroup$ For the interpretation that no vowel is next to another vowel, that would have a result of $6!\times 7\times 6\times 5\times 4 = 840\times 6!$, which doesn't match any of the options either. $\endgroup$ – JMoravitz Mar 13 at 17:14
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    $\begingroup$ @BJKShah which, if you would notice $\dfrac{7!}{4!3!}\cdot 4!\cdot 6! = 604800=6!\cdot 7\cdot 6\cdot 5\cdot 4$ so your answer is the same as the one I gave last week. $\endgroup$ – JMoravitz Mar 18 at 12:02
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    $\begingroup$ It is the same idea as yours, except rather than arranging the vowels in a row outside, and then inserting them into the simultaneously picked holes after having arranged, I instead inserted them into the holes one at a time. Worded again another way, $6!$ to arrange the consonants. Then $7$ choices for where the a goes, $6$ choices for where the e goes, etc... $\endgroup$ – JMoravitz Mar 18 at 12:20

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