Q.Letters of the word MOTHER are arranged at random. Find the probability that in the arrangement a) vowels are always together b) vowels are never together c) $O$ is at the beginning and end with $T$ d) starting with a vowel and end with a consonant.

$n(S)=6!=720$.

a) Vowels together: glue O,E into one block (2! internal orders), plus 4 consonants = 5 units in $5!$ orders: $5!\times2!=240$, $P=240/720=1/3$.

b) Vowels never together is the complement: $P=1-1/3=2/3$.

c) Starts with O, ends with T: the other 4 letters fill the middle in $4!=24$ ways, $P=24/720=1/30$. …

Factorial arrangement counting: block together items that must be adjacent; fix specific positions and arrange th …

Forgetting the internal $2!$ arrangement of O and E within their block in part ( …