Question 467690: how many ways can you arrange A-F with a consonant at one end, and a vowel at the other?
we answered other similiar problems using factorials
Found 2 solutions by ccs2011, sudhanshu_kmr:
Answer by ccs2011(207)   (Show Source):
You can put this solution on YOUR website! **For any set of n objects, there are n! ways of arranging them**
Now for the letters A-F, there are 2 vowels and 4 consonants
If a vowel is at one end and a consonant at the other then that leaves 4 letters remaining to be arranged in any which way between them.
From above we know then that are 4! ways of arranging 4 letters.
Say the consonant B is on the front end and a vowel on other end:
B _ _ _ _ Vowel
There are only 2 possibilities because there are only 2 vowels to choose from:
B _ _ _ _ A OR B _ _ _ _ E
Each of these has 4! arrangements
So with B at the front there are 2*(4!) arrangements
However, there are 4 consonants each of which could be at the front end
So multiply by 4
With any consonant at front end there are 4*2*(4!) arrangements
However, the consonant could also be at back end with vowel in front
Since everything else would stay the same, double the possible arrangements
Total number of arrangements are 2*4*2*(4!) arrangements which simplifies to 16(4!)
Answer by sudhanshu_kmr(1152)  (Show Source):
You can put this solution on YOUR website!
A to F, total 6 letters where 2 vowels and 4 consonants.
Number of ways when vowel at first and one consonant at last: 2*4*4! = 192
similarly ,
number of ways when vowel at last and one consonant at first: 4*2*4! = 192
total number of ways = 192 + 192 = 384
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