\\ $Consider the words APPLES and ORANGES. Let the number of distinct permutations of the letters in the word APPLES be $ P_A $ and similarly $ P_O $ for oranges.$ \\\\ $a) By considering the ratio $ \frac{P_O}{P_A} $, prove that $ P_O > P_A . \\\\ $b) Deduce that oranges are greater than apples.$