Probability of rolling higher than $N$ by summing the highest $X$ number of dice out of a set $Y$ number of dice, each with $Z$ sides.

Question

I'd like help finding a formula for the probability of rolling higher than a target number, $N$, by summing the highest $X$ number of dice out of a set $Y$ number of dice, each with $Z$ sides, numbered $1 \to Z$.

Given (and obvious): $N < X*Z, \:\:X < Y$

Edit #1: For sake of simplicity lets make $Z=6$ and $X=3$

In dice notation it would be $P(Yd6k3>N)$. Thanks to SuperJedi224 for that.

I've tried brute force through an Excel sheet and http://rumkin.com/reference/dnd/diestats.php but the Excel sheet got really cumbersome after 46656 lines ($6d6k3$) and the website bugged out at $8d6k3$.

Edit #2: What would the odds be of each of the possible outcomes on $Y$ dice?

For example, what are the odds that $Y$ dice roll the set (4,5,3)? If that could be figured out for each of the $6^3$ possible outcomes, it would get us really close to an answer for the original question.