<PRE><B>Question 360325</B>
without any restrictions, the number of possible ways to arrange the letters VIOLENT would be 7 * 6 * 5 * 4 * 3 * 2 * 1 = 7! = 5040


this can only happen in one way.


this assumes that each letter can only be used once.


since there are 3 vowels and 4 consonants, and the vowels can only be in odd positions, then the possible ways this can happen are provided by the following formula:


number of possible ways to get sets of 3 out of a total of 4 is:


4! / (3! * 1!) = (4*3!)/3! = 4


the general equation for this is:


n! / (x! * (n-x)!)


n = 4
x = 3
(n-x) = 1


the number of combinations in each one of these ways is equal to 4! * 3! = 144


since there are 4 ways that can happen as we just calculated above, then the total ways you can get 3 vowels in only the odd position is 4 * 144 = 576.


the 4 possible positions the vowels can be in are shown as follows:


x shows the position of the vowel.
c shows the position of the consonant.


&lt;pre&gt;

position          1   2   3   4   5   6   7
way number 1      x   c   x   c   x   c   c
way number 2      x   c   x   c   c   c   x
way number 3      x   c   c   c   x   c   x
way number 4      c   c   x   c   x   c   x

&lt;/pre&gt;


the number of possible combinations in each of the ways is calculated as follows:


we&#39;ll take way number 1 to show you. 


the other ways work the same.


way number 1


vowel in 1st and 3d and 5th position.


you have 3 vowels.


the first vowel position can take any 3 of them.
the second vowel position can take any 2 of them.
the third vowel position can take any 1 of them.


total possible combinations are therefore 3*2*1 = 6 = 3! for the vowels.


you have 4 consonants.


the first consonant position can take any 4 of them.
the second consonant position can take any 3 of them.
the third consonant position can take any 2 of them.
the fourth consonant position can take any 1 of them.


total possible combinations are therefore 4*3*2*1 = 24  = 4! for the consonants.


for each possible combination of vowels, there are 24 possible combinations of consonants.


since there are 6 possible combinations of vowels, then the total number of possible combinations of vowels and consonants for way number 1 is 6 * 24 = 144.


4 possible ways means 4 * 144 = 576.












</PRE>