How many words can be formed with the letters of SCIENCE so that the vowels are not together.
There are three vowels in the word, SCIENCE,
Let's find how we can arrange these letters in different combinations first and find how we can arrnage these letters so that the vowels are together. The difference between those two is the number of ways that we can arrange these letters so that vowels are not together.
First, we shall find the total number of different combinations, In this seven letter word both C and E are repeating twice. So the total number of cominations would be,
`T = (7!)/(2!2!) = 1260`
Therefore there are 1260 different ways of arranging these seven letters,
Now let's arrange this so that all the vowels are together,
SCNC(IEE) we will treat the (IEE) as one letter.
So there are five letters and C is repeating twice.
So the the number of arrnagements with vowels are together is,
`V = (5!)/(2!) = 60`
but our special letter (IEE) itself can be aranged inside to give different combinations. The three letters IEE can be arranged inside the bracket in `(3!)/(2!) = 3` ways.
Therefore the total number of combinations with vowels are together is Tv
`Tv = 60 * 3 = 180`
Therefore the total number of arrangements with vowels are not together is (T -Tv),
T-Tv = 1260 - 180 = 1080
So there are 1080 ways that we can arrange these 7 letters without vowels are not together.