Suppose that instead of swapping element $A[i]$ with a random element from the subarray $A[i \dots n]$, we swapped it with a random element from anywhere in the array:

PERUMTE-WITH-ALL(A)

1 n = A.length

2 for i = 1 to n:

3     swap A[i] with A[RANDOM(1, n)]

Does this code produce a uniform random permutation? Why or why not?

This code does not produce a uniform random permutation.

Unlike the code in Exercise 5.3-2, this code can actually produce all random permutations. The trouble comes from the uniform production requirement. Consider the case when $n = 3$. We would expect all 6 possible permutations to be equally likely. Within the for loop we are exchanging $A[i]$ with one of 3 different possible values, meaning there are $3^3 = 27$ possible non-distinct outputs each with a $\frac{1}{27}$ chance of ocurring. No matter how the possibilities are distributed, it is not possible to form a $\frac{1}{6}$ chance by repeatedly summing $\frac{1}{27}$ and so the production cannot be uniform.