A10 ohm resistor is connected in parallel across series combination of 4 ohms and 6 ohms resistors. calculate the amount of heat produced per second, in 4 ohms resistor if 10 joules heat is produced per second in 10 ohms resistor

OK.  We can do this  !

--  We need to know the current through the 4-ohm resistor.

-- But the only thing we know is the power dissipated by the 10-ohm
resistor.  Can we use that to find the current through the 4-ohm resistor ?

You bet we can !

-- The effective resistance of the series combination  is (4+6) = 10 ohms.

-- Both parallel branches have the same resistance, so the current
is the same in both branches.

-- The power dissipated by a resistor is          P = I² · R
For the 10-ohm resistor,

             (10 joules/sec) = (I²) · 10-ohms

                   I = √(10/10)  =  1 ampere

-- The current in the series combination is the same as the current
in the 10-ohm resistor . . . 1 ampere.

-- So the power dissipated by the 4-ohm resistor in the series branch is

                 P = I² · R  =  (1 amp)² · (4 ohms)  =  4 watts.

Ohm's law tells you that i = v/r where v is equivalent voltage and r is equivalent resistance of the circuit
since all circuits have the same voltage, your question is really asking which circuit has the lowest total resistance, since the lower the resistance, the higher the current
circuit a has a resistance of 5 ohms; circuit c is in series so has a resistance of 10 ohms; circuit b is in parallel and therefore has a resistance of 2.5 ohms (review your notes on adding resistances in parallel and series); therefore b has the highest current