How many words can be formed with the letters of the word 'CALCULUS'? Out of these words, how many words
(i) starts with A ?
(ii) starts with A and ends with S?
(iii) have all vowels together?
(iv) have not all vowels together?

Explanation

Total letters in the word 'CALCILUS' = 8, in which C,L and U occur twice and A and S occur once.
Total words `= (8!)/(2!.2!.2!) = 5040`.
(i) For the words starting with A
Remaining letters = 7
Here C,L,U occur twice.
Therefore, total words starting with A `= (7!)/(2!.2!.2!) = 630`.
(ii) For the words starting with A and ending with S Remaining letters = 6
Here C,L,U occur twice
Therefore, total words starting with A and ending with `S = (6!)/(2!.2!.2!) = 90`.
(iii) Vowels = A,U,U
Consonants = C,C,L,L,S
Treating all vowels as one letter Total letters = 6
In which C and L occur twice.
Now number of arrangements of 6 letters at 6 places `= (6!)/(2!.2!) = 180`
No. of arrangements of 3 vowels (A,U,U) are 3 places `= (3!)/(2!) = 3`
No. of arrangements of 3 vowels (A,U,U) ar 3 places `= (3!)/(2!) = 3`
`:.` Total words in which all vowels occur together. `= 180 xx 3 = 540`.
(iv) Number of words in which vowels never occur together
= Total words - No. of words in which vowels are always together
`= 5040 - 540 = 4500`