Two fair cubical dice, find the values of $t$ Two fair cubical dice are thrown simultaneously and the scores multiplied. $P(n)$ denotes the probability that the number $n$ will be obtained. If $P(t)=\frac{1}{9}$, find the possible values of $t$.

There are $36$ possible dice rolls, so there must be $4$ ways to obtain $t$ for $P(t)$ to be $\frac19$. In other words, $t$ is a number that can be written as a product of two numbers between $1$ and $6$ in exactly four ways. We can quickly see that the only such numbers are: $$t=12=2\cdot6=3\cdot4=4\cdot3=6\cdot2$$

or $$t=6=1\cdot6=2\cdot3=3\cdot2=6\cdot1$$