Calculate the number of ways to arrange all the letters of the word LONGITUDES if all the vowels are separated from each other .
i did this , 10!-(7!x4!) but its wrong .
Dear thereddevils,
First you could seperate the consonants L N G T D S. These six letters could be arranged in 6! different ways.Then the remaining vowels could be put inbetween these letters. (only one vowel must be put between two letters) To do this there are, $\displaystyle _{4}^{7}\textrm{P}$ ways (there are four vowels) since there are seven vacant places around the letters L N G T D S.
Therefore the number of ways to arrange all the letters= $\displaystyle {6!}\times_{4}^{7}\textrm{P}=604800$
If you have any problem with the above method please don't hesitate to reply me.