So far I have worked out the total number of combinations with no repeats which I believe to be
10 x 9 x 8 x 7 = 5040
I'm not quite sure where to proceed after this, I think I have to work out the total number of combinations with a 1 and 0 in it and then subtract this from 5040 for my answer. However, i'm not sure how to work this out. Any help would be appreciated. Thanks!
Assuming the codeword can start with $0$
As you have done, total number of combinations with no repeats $=10 \times 9 \times 8 \times7 =5040$
To find the number of codewords with both $0$ and $1,$ include both of them and select other $2$ digits in $\dbinom{8}{2}$ ways. these $4$ digits can be arranged in $4!$ ways. So, $\dbinom{8}{2} \times 4!=672$ ways.
Thus you get $5040-672=4368$ codewords