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# Problem 480

The Last Question

Consider all the words which can be formed by selecting letters, in any order, from the phrase:

Suppose those with 15 letters or less are listed in alphabetical order and numbered sequentially starting at 1.
The list would include:

1 : a
2 : aa
3 : aaa
4 : aaaa
5 : aaaaa
6 : aaaaaa
7 : aaaaaac
8 : aaaaaacd
9 : aaaaaacde
10 : aaaaaacdee
11 : aaaaaacdeee
12 : aaaaaacdeeee
13 : aaaaaacdeeeee
14 : aaaaaacdeeeeee
15 : aaaaaacdeeeeeef
16 : aaaaaacdeeeeeeg
17 : aaaaaacdeeeeeeh

28 : aaaaaacdeeeeeey
29 : aaaaaacdeeeeef
30 : aaaaaacdeeeeefe

115246685191495242: euleoywuttttsss
115246685191495243: euler
115246685191495244: eulera

525069350231428029: ywuuttttssssrrr

Define P(w) as the position of the word w.
Define W(p) as the word in position p.
We can see that P(w) and W(p) are inverses: P(W(p)) = p and W(P(w)) = w.

Examples:

W(10) = aaaaaacdee
P(aaaaaacdee) = 10
W(115246685191495243) = euler
P(euler) = 115246685191495243

Find W(P(legionary) + P(calorimeters) - P(annihilate) + P(orchestrated) - P(fluttering)).

1 : a
2 : aa
3 : aaa
4 : aaaa
5 : aaaaa
6 : aaaaaa
7 : aaaaaac
8 : aaaaaacd
9 : aaaaaacde
10 : aaaaaacdee
11 : aaaaaacdeee
12 : aaaaaacdeeee
13 : aaaaaacdeeeee
14 : aaaaaacdeeeeee
15 : aaaaaacdeeeeeef
16 : aaaaaacdeeeeeeg
17 : aaaaaacdeeeeeeh

28 : aaaaaacdeeeeeey
29 : aaaaaacdeeeeef
30 : aaaaaacdeeeeefe

115246685191495242: euleoywuttttsss
115246685191495243: euler
115246685191495244: eulera

525069350231428029: ywuuttttssssrrr

W(10) = aaaaaacdee
P(aaaaaacdee) = 10
W(115246685191495243) = euler
P(euler) = 115246685191495243