Getting unique combination of elements from lists within a list

Question

I have this list:

list = [['a','b'],
        ['1', '2', '3'],
        ['r', 's', 't','u']]

I want to have all the unique combinations of all elements from the sub-list in a list like bellow:

['a1r','a1s','a1t','a1u','a2r','a2s','a2t','a2u','a3r','a3s','a3t','a3u',
'b1r','b1s','b1t','b1u','b2r','b2s','b2t','b2u','b3r','b3s','b3t','b3u']

I have tried writing this code, which only gives me first four combinations:

list = [['a','b'],
        ['1', '2', '3'],
        ['r', 's', 't','u']]
i = j = k =  0
x = y = z = ''
str = ''
final_list = []
while i <= 1:
    x = list[0][i]
    i += 1
    while j <= 2:
        y = list[1][j]
        j += 1
        while k <= 3:
            z = list[2][k]
            k += 1
            str = x + y + z
            final_list.append(str)

print(final_list)

The output I get from my code is:

['a1r', 'a1s', 'a1t', 'a1u']

What changes should I make?