D A U G H T E R
1 2 3 4 5 6 7 8
[Notice that DAUGHTER itself is NOT one of the
required permutations because A is in position 2,
which is not an odd position]
There are 4 odd positions: 1,3,5,7.
There are 3 vowels: A,E,U
An example of one of the desired permutations would
be RTEHAGUD,
R T E H A G U D
1 2 3 4 5 6 7 8
Take away the consonants and leave only the
vowels for illustration:
E A U
1 2 3 4 5 6 7 8
The permutation of odd positions chosen here
for A,E,U were 5,3,7 respectively.
We placed the A in position 5, the E in position 3,
and the U in position 5.
E A U
1 2 3 4 5 6 7 8
There are 4P3 permutations of 3 of the odd positions 1,3,5,7
The one that was chosen in this example was 5,3,7.
In that example, We placed A in the odd position 5 that
came first in the PERMUTATION 5,3,7 of odd positions,
(not that came first in the word!).
Then we placed E in the odd position that came second
in the PERMUTATION 5,3,7 of odd positions, (not that
came second in the word!).
Finally we placed U in the odd position that came
third in the PERMUTATION 5,3,7 of odd positions, (not that
came third in the word!)
There are 5 remaining positions in which to place the 5
remaining letters which is 5P5.
Answer: (4P3)(5P5) = 24(120) = 2880.
Edwin
