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Mary, Mary, Quite Orderly (Posted on 2014-09-24) Difficulty: 3 of 5
Determine the total number of ways the letters of the word supercalifragilisticexpialidocious can be arranged such that:

(i) All the five vowels will occur in order.
(ii) At least four of the five vowels will occur in a precisely reversed order.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts re(3): solution -- corrected part (ii) it is hoped | Comment 4 of 6 |
(In reply to re(2): solution by Charlie)


As Jer has pointed out, the treating of a given vowel as if it were a consonant will include not only disorderings of that consonant, but also, by happenstance, all the intended orderings (i.e., reversed) as well. Also, among the disorderings, interchange of two adjacent vowels' positions will be counted twice: once with each of the two vowels in question.

By the first of these two factors that were overlooked, instead of adding in the ordered case (16331695263739215360000 ways), we should have subtracted it out 4 times (equivalently subtracting it out once for each of the 5 vowel conditions, then adding it in for itself).

The second of these two types of previously overlooked condition involves the double counting of four specific orderings. As they are specific orderings, they also each have the same number of ways: 16331695263739215360000.

Bottom line, it seems we should take the sum of the five left-out-vowel conditions and subtract eight times the specific-order count:

  9145749347693960601600000 + 1959803431648705843200000 + 186834593817176623718400000 +  1959803431648705843200000 + 1959803431648705843200000 - 8 * 16331695263739215360000

=  201729099897706788126720000 

  Posted by Charlie on 2014-09-25 09:42:30
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