1 inch x 1 inch tiles

There are $9$ letters in the word "BRILLIANT" .

Two letters, "I" and "L" can be picked twice. So, the number of favorable outcomes is $2$ .

The total number of ways of picking two letters from the word "BRILLIANT" are

$\Large{9 \choose 2}$ $=$ $\dfrac{9\cdot8}{2}$ $=36$

So the probability that the two letters are the same letters is Number of favorable outcomes $\div$ Total number of Outcomes

$=$ $\dfrac{2}{36}$

$=\dfrac{1}{18}$ .

Since $1$ and $18$ are co-prime,

$\dfrac{x}{y}=\dfrac{1}{18}$ .

$\implies x=1, y=18$ .

$3x^{321} +3y^3$

$=3\cdot1^{321}+3\cdot18^3$

$=3\cdot1 +3\cdot5832$

$=3 + 17496$

$=\boxed{17499}$