This answer is only true if you disregard the rules that the mastermind tag implies. Nothing except the tag hints that masterminds rules should apply here and usually when these things are passed around they do not come with Masterminds rules included. That tag was also not put there by the person asking the question, so I will look at only what is written on that piece of paper.
In my opinion this puzzle has technically 13 possible solutions and each one of those is equally valid.
The statements, lets call them a, b, c and d:
a - 2 6 5 7 - Has two correct digits but neither are in the correct place
b - 0 4 1 5 - Has one correct digit but it's in the wrong place
c - 4 2 6 8 - Has no correct digits
d - 1 7 4 9 - Has two correct digits, both in the correct place
Starting with c we know that 4 2 6 8 are not in the correct combination, so valid digits are 013579, lets call them X. There are 6 possible digits so we have 6*6*6*6=1296 possible combinations:
XXXX
By combining c and d we know that 4 can not be in the solution. And given that two numbers are in the correct position we can narrow the possibilities down by quite a bit. The only possible options now are:
XXXX
17XX, 1XX9, X7X9 (possible combinations of two numbers being in correct positions, excluding 4)
In total this is 6*6+6*6+6*6=108 possible solutions with these two rules.
Adding the statement a narrows it down further. 2 and 6 can not be in the solution (statement c), so 5 and 7 must be in the solution and as they are presented they are in the wrong place.
XXXX
17XX, 1XX9, X7X9
17X5, 1579, 57X9 (5 can not be in the third position)
This brings the number of possible solutions down to 6+1+6=13
This is also as low as I think it is possible to get with the given input, statement b is true regardless of the value of X. All of the 13 possible combinations of 17X5, 57X9 and 1579 where X is [0, 1, 3, 5, 7, 9] satisfy all of the statements.
I will take X=0 with both combinations that contain X for the examples(1705, 5709). Parts of the number that satisfies each statement is in bold.
a - 2 6 5 7 - Has two correct digits but neither are in the correct place
1705, 5709
b - 0 4 1 5 - Has one correct digit but it's in the wrong place
1705 , 5709 Having two improperly placed correct numbers and only stating one contradicts masterminds rules, but does not contradict the fact, that a correct digit is out of place.
c - 4 2 6 8 - Has no correct digits
1705, 5709
d - 1 7 4 9 - Has two correct digits, both in the correct place
1705, 5709
The only line where the variables X is highlighted is with statement b and that statement is satisfied regardless of what value the X has.