This comment was posted to reddit on Jan 28, 2016 at 7:08 am and was deleted within 9 minutes.

If a given state can be accomplished in more ways, then it is more probable than the state that can only be accomplished in a fewer/one way. Assume a box filled with jigsaw pieces were jumbled in its box, the probability that a jigsaw piece will land randomly, away from where it fits perfectly, is very high. Almost every jigsaw piece will land somewhere away from its ideal position. The probability of a jigsaw piece landing correctly in its position, is very low, as it can only happened one way. Thus, the misplaced jigsaw pieces have a much higher multiplicity than the correctly placed jigsaw piece, and we can correctly assume the misplaced jigsaw pieces represent a higher entropy. Derivation and Explanation To understand why entropy increases and decreases, it is important to recognize that two changes in entropy have to considered at all times. The entropy change of the surroundings and the entropy change of the system itself. Given the entropy change of the universe is equivalent to the sums of the changes in entropy of the system and surroundings: ΔSuniv=ΔSsys+ΔSsurr=qsysT+qsurrT(1) (1) Δ S u n i v Δ S s y s Δ S s u r r q s y s T q s u r r T In an isothermal reversible expansion, the heat q absorbed by the system from the surroundings is qrev=nRTlnV2V1(2) (2) q r e v nRTln V 2 V 1 Since the heat absorbed by the system is the amount lost by the surroundings, qsys=−qsurr q s y s q s u r r .Therefore, for a truly reversible process, the entropy change is ΔSuniv=nRTlnV2V1T+−nRTlnV2V1T=0(3) (3) Δ S u n i v n R T ln V 2 V 1 T n R T ln V 2 V 1 T 0 If the process is irreversible however, the entropy change is ΔSuniv=nRTlnV2V1T>0(4) (4) Δ S u n i v n R T ln V 2 V 1 T 0 If we put the two equations for ΔSuniv Δ S u n i v together for both types of processes, we are left with the second law of thermodynamics, ΔSuniv=ΔSsys+ΔSsurr≥0(5) (5) Δ S u n i v Δ S s y s Δ S s u r r 0 where ΔSuniv Δ S u n i v equals zero for a truly reversible process and is greater than zero for an irreversible process. In reality, however, truly reversible processes never happen (or will take an infinitely long time to happen), so it is safe to say all thermodynamic processes we encounter everyday are irreversible in the direction they occur.