More example:
* In how many different ways can the letters of the word ‘OPTICAL’ be arranged so that the vowels always come together?
The word ‘OPTICAL’ contains 7 different letters.
When the vowels OIA are always together, they can be supposed to form one letter.
Then, we have to arrange the letters PTCL (OIA).
Now, 5 letters can be arranged in 5! = 120 ways.
The vowels (OIA) can be arranged among themselves in 3! = 6 ways.
Required number of ways = (120 x 6) = 720.
** In how many different ways can the letters of the word ‘MATHEMATICS’ be arranged so that the vowels always come together?
In the word ‘MATHEMATICS’, we treat the vowels AEAI as one letter.
Thus, we have MTHMTCS (AEAI).
Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.
Number of ways of arranging these letters = 8!/{ (2!)(2!)}
= 10080.
Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.
Number of ways of arranging these letters
= 4!/2!= 12.
So, Required number of words = (10080 x 12) = 120960.
এই লেকচারের পরের পেইজে যেতে নিচের 
…. তে ক্লিক কর।