Say we have a $k$-sided die, and we want to throw it $n$ times.

What's the expression for the probability of getting any one face up at least $m$ times, where $m \in \{0, ... , n\}$ and $x \in \{1, ... , k\}$?

Edit: The question previously asked about the probability of getting a specific face at least $m$ times. I realized that's not what I wanted, my apologies. I'm instead looking for the probability of getting any one face at least $m$ times. So whether we get '1' at least $m$ times, or '2' at least $m$ times, or '$k$' at least $m$ times, they will all be favorable outcomes.

Say we have a $k$-sided die, and we want to throw it $n$ times.

What's the expression for the probability of getting any one face up at least $m$ times, where $m \in \{0, ... , n\}$?

Edit: The question previously asked about the probability of getting a specific face at least $m$ times. I realized that's not what I wanted, my apologies. I'm instead looking for the probability of getting any one face at least $m$ times. So whether we get '1' at least $m$ times, or '2' at least $m$ times, or '$k$' at least $m$ times, they will all be favorable outcomes.

Say we have a $k$-sided die, and we want to throw it $n$ times.

What's the expression for the probability of getting the $x$ face up at least $m$ times, where $m \in \{0, ... , n\}$ and $x \in \{1, ... , k\}$?

Say we have a $k$-sided die, and we want to throw it $n$ times.

What's the expression for the probability of getting any one face up at least $m$ times, where $m \in \{0, ... , n\}$ and $x \in \{1, ... , k\}$?

Edit: The question previously asked about the probability of getting a specific face at least $m$ times. I realized that's not what I wanted, my apologies. I'm instead looking for the probability of getting any one face at least $m$ times. So whether we get '1' at least $m$ times, or '2' at least $m$ times, or '$k$' at least $m$ times, they will all be favorable outcomes.