Q.The letters of the word 'EQUATION' are arranged in a row. Find the probability that a) All the vowels are together b) Arrangement starts with a vowel and ends with a consonant.

$n(S)=8!=40320$ (all letters of EQUATION - E,Q,U,A,T,I,O,N - are distinct).

a) Vowels (E,U,A,I,O - 5 of them) together: glue them into one block ($5!$ internal orders) plus 3 consonants = 4 units in $4!$ orders: $4!\times5!=24\times120=2880$, $P=2880/40320=1/14$.

b) Starts with a vowel (5 choices), ends with a consonant (3 choices), remaining 6 letters fill the middle in $6!=720$ ways: $5\times3\times720=10800$, $P=10800/40320=15/56$.

Factorial arrangement counting: block the vowels together for part (a); fix the first and last letter categories and arrange the rest for part (b).

Forgetting the internal $5!$ arrangement of the vowels within their block in part (a).