If two letters are taken at random from the word HOME, what is the probability that none of the letters would be vowels?

P(first letter is not vowel) =  $\frac{2}{4}$

P(second letter is not vowel) =  $\frac{1}{3}$

So, probability that none of letters would be vowels is =  $\frac{2}{4}\times \frac{1}{3}=\frac{1}{6}$