Suppose that instead of swapping element $A[i]$ with a random element from the subarray $A[i \ldots n]$, we swapped it with a random element from anywhere in the array:
PERMUTE-WITH-ALL(A) n = A.length for i = 1 to n swap A[i] with A[RANDOM(1,n)]
Does this code produce a uniform random permutation? Why or why not?
It does not. Intuitivelly, this one can go in $n^n$ different ways while there are $n!$ combinations. Since $n!$ does not divide $n^n$, there is no way that this can be a uniform distribution. (Why doesn't it divide $n^n$? That's the intuitive part. $n!$ is divisable by $n-1$, but $n^n$ can't be for $n > 2$).
Of course, this is a popular problem and there are tons of posts and papers written on it. Here's one from Coding Horror