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 Mary, Mary, Quite Orderly (Posted on 2014-09-24)
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 No Rating

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 solution | Comment 1 of 6
SUPERCALIFRAGILISTICEXPIALIDOCIOUS contains:

`3 a's v0 b's3 c's1 d's2 e's v1 f's1 g's0 h's7 i's v0 j's0 k's3 l's0 m's0 n's2 o's v2 p's0 q's2 r's3 s's1 t's2 u's v0 v's0 w's1 x's0 y's0 z's`

That's 16 vowels and 18 consonants.

For Part (i):

As far as vowels go, the order is given, so we're concerned only with which 16 positions out of the 34 are used by the vowels: C(34,16) =  2203961430.

In the remaining 18 positions, we are concerned with the order, but there are duplicates, within which we don't care which P comes first, for example. The 18 consonants have only ten identities, found in quantities of 3, 1, 1, 1, 3, 2, 2, 3, 1 and 1. So the consonants multiply the vowel result by 18! / (3! * 3! * 2 * 2 * 3!) =  7410154752000.

The answer for part (i) is then C(34,16) * 18! / (3! * 3! * 2 * 2 * 3!) = 2203961430 * 7410154752000 = 1633169526373921560000.

For Part (ii):

Calculation is done similarly to the above, but six cases need to be considered: all except A are in reverse order, all except E, etc, and all five are in reverse order. The cases are then added.

The simplest is that all five are in reverse order: that's the same as the answer to part (i) as it's still a specified order.

In the other cases, the vowel left out of the reverse order is treated just like a consonant.

A: C(34,13) * 21! / (3! * 3! * 3! * 2 * 2 * 3!) = 9145749347693960601600000

E: C(34,14) * 20! / (2 * 3! * 3! * 2 * 2 * 3!)

I: C(34,9) * 25! / (7! * 3! * 3! * 2 * 2 * 3!)

O: C(34,14) * 20! / (2 * 3! * 3! * 2 * 2 * 3!)

U: C(34,14) * 20! / (2 * 3! * 3! * 2 * 2 * 3!)

The six numbers to be added are:

`    16331695263739215360000  9145749347693960601600000  1959803431648705843200000186834593817176623718400000  1959803431648705843200000  1959803431648705843200000`

The total is 201876085155080441064960000 to answer part (ii).

Note that in treating a given vowel as a consonant, it can be interspersed with the other vowels, which I assume is what is allowed by the rules. That is, it need not be in successive vowel positions, so that successive vowels might be, for example, ignoring the consonants: UUOAOIIIAIAIIIEE.

 Posted by Charlie on 2014-09-24 16:00:59

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