# Probability of getting maximum dots in n rolls of a die

A fair die is rolled $n$ times. Let $X_i$ denote the number of spots that are on the up face on roll $i$, for $i=1, 2, \dots, n$. Find the probability mass function of $Y=\text{max}\{X_1,X_2,\dots,X_n\}$.

I've been trying various things but I know my answers aren't correct because the sum of the values of the pmf don't equal $1$.

The probability that the maximum $Y$ is $\le k$ is the probability that all the $X_i$ are $\le k$. This is $\left(\frac{k}{6}\right)^n$.
The probability that $Y=k$ is the probability that $Y$ is $\le k$ minus the probability that $Y$ is $\le k-1$.