Well its very simple.
See first lets calculate the sample space. It would be
$$
{^{11}C_4}
$$
because you are choosing a total of 4 balls out of 11 balls.
Next you need to calculate the favorable event
so you have 6 blue balls and favorable would be when you take out 4 blue ones.
which can be done by
$$
{^6C_4}
$$
ways. So the probability becomes
$$
={{^6C_4}\over{^{11}C_4}} \\={1\over 22}
$$
If you dont get it by this method here is the basic step by step trick:-
Probability of getting first blue ball is
$$
={6\over 11}
$$
as there are total 11 and 6 blue
now Probability of getting Second blue ball is
$$
={5\over 10}
$$
because you took out 1 ball from total which is blue so both numbers reduce by 1.
similarly
3rd blue ball
$$
={4\over 9}
$$
and 4th one
$$
={3\over 8}
$$
so final probablity is
$$
P =\frac{6}{11} \cdot \frac{5}{10}\cdot \frac{4}{9}\cdot \frac{3}{8}\\P=\frac{360}{7920}\\P=\frac{1}{22}\\
$$