How many students failed in Physics and Maths ?

Number of students = 280

Students who passed in Maths = $$m + x + w + y = 185$$ ----------Eqn(1)

Students who passed in Physics = $$p + x + w + z = 210$$ --------Eqn(2)

Students who passed in Chemistry = $$c + y + w + z = 222$$ ------------Eqn(3)

Now, Students who passed in Maths and Physics = $$x + w = 185 - 5 = 180$$ ------------Eqn(4)

Students who passed in Maths and Chemistry = $$w + y = 185 - 10 = 175$$ ------------Eqn(5)

Students who passed in Physics and Chemistry = $$z + w = 200$$ ------------Eqn(6)

Number of Students who passed in at least one subject = $$m + p + c + w + x + y + z = 280 - 47 = 233$$ ------------Eqn(7)

Now, adding equations (1),(2) and (3) and subtracting eqn(7) from it, we get :

=> $$x + y + z + 2w = 384$$ ------------Eqn(8)

Adding Eqn (4),(5) and (6)

=> $$x + y + z + 3w = 555$$ ------------Eqn(9)

Now, equation (9)-(8)

=> $$w = 171$$

Putting it in equation (4), we get $$x = 9$$

Similarly, solving all equations , the required value of the variables are :

Number of students who failed in Physics and Maths = students who passed only in chemistry + students who failed in all subjects

= $$18 + 47 = 65$$