Words are formed using all the letters of the word W = POTATO. Let 'a' denotes the number of words when all the vowels are together 'b' denotes the number of words when all the vowels as well as all the consonants are separated from each other and 'c' denotes the number of words which begin and end in vowels. Which of the following is (are) correct?

a' denotes the number of words when all the vowels are together

(OAO),P,T,P

$a=\frac{4!}{2!}\times \frac{3!}{2!}=12\times 3=36$

b' denotes the number of words when all the vowels as well as all the consonants are separated

Vowels O,A,O can be arranged in $\frac{3!}{2!}$ways.

consonants P,T,T can be arranged in $\frac{3!}{2!}$ways

(OAO),(PTT) can be rearranged in 2! ways

$b=\frac{3!}{2!}\times \frac{3!}{2!}\times 2!=18$

c' denotes the number of words which begin and end in vowels. 

$c=\frac{4!}{2!}+\frac{4!}{2!}\times 2!=12+24=36$

$a=36;b=18;c=36$