Rand v0.6.0

The Rng::shuffle method is now deprecated; rand::seq::SliceRandom trait should be used. It provides the shuffle() method on all slices, which accepts an Rng instance:

// Rust edition 2018 no longer needs extern crate

use rand::thread_rng;
use rand::seq::SliceRandom;

fn main() {
    let mut vec: Vec<u32> = (0..10).collect();
    vec.shuffle(&mut thread_rng());
    println!("{:?}", vec);
}

See it on Playground.

Original answer

You're very close. This should work:

extern crate rand;

use rand::{thread_rng, Rng};

fn main() {
    let mut vec: Vec<u32> = (0..10).collect();
    let slice: &mut [u32] = &mut vec;

    thread_rng().shuffle(slice);
}

&mut [T] is implicitly coercible to &[T], and you annotated the slice variable with &[u32], so the slice became immutable: &mut [u32] was coerced to &[u32]. mut on the variable is not relevant here because slices are just borrows into data owned by someone else, so they do not have inherited mutability - their mutability is encoded in their types.

In fact, you don't need an annotation on slice at all. This works as well:

extern crate rand;

use rand::{thread_rng, Rng};

fn main() {
    let mut vec: Vec<u32> = (0..10).collect();
    let slice = vec.as_mut_slice();

    thread_rng().shuffle(slice);
}

You don't even need the intermediate variable:

extern crate rand;

use rand::{thread_rng, Rng};

fn main() {
    let mut vec: Vec<u32> = (0..10).collect();
    thread_rng().shuffle(&mut vec);
}

You should read The Rust Programming Language as it explains the concepts of ownership and borrowing and how they interact with mutability.