For coin, assume heads (H) represents ordered state & tails (T) represents random distribution.
If coin tossed, there are 2 ways to represent:
1 ordered state (H) & 1 random distribution (T)

If 2 coins tossed, there are 4 ways to represent (2²):
HH  HTTHTT  (1 ordered state (HH) & 3 random distributions)

If 3 coins tossed, there are 8 ways to represent (2³):
HHH  HHTHTHTHHHTTTHTTTHTTT
(1 ordered  &  7  random distributions)
 

Consider box of 20 quarters shown:

There are 2²⁰ or 1,048,576 ways to represent
(assume 1 ordered state: HHHHHHHHHHHHHHHHHHHH)

Arrange the 20 quarters in ordered state and shake. The system (box) moves spontaneously to random distribution (more ways to represent).

Ludwig Boltzmann was first to really understand entropy.
In 1896, he proposed:
S = k (ln W)      k = Boltzmann constant
                                       W= No. ways state can be achieved

Assume the 20 quarters has 2 states:
1. Ordered (20H)
2. Random distribution (all other combinations)

Entropy for ordered state (W=1):
S = k (ln W) = k ln 1 = 0

Entropy for random distribution (W=2²⁰):
S = k (ln W) = (1.38x10^-23 J/K)(ln 2²⁰) =
                     = (1.38x10^-23 J/K)(20 ln2) = 1.91x10^-22 J/K (higher entropy)

Consider 1 mol of quarters and assume 2 states:
1. Ordered (6x10²³H)
2. Random distribution (all other combinations)
 
 

Entropy of ordered state:
S = k (ln W) = k ln 1 = 0     (only 1 way to achieve)

Entropy for random distribution:
There are 2^6E23 ways to achieve (W=2^6E23)
S = k (ln W) = (1.38x10^-23 J/K)(6x10²³ ln 2) = 5.7 J/K
 
 

Consider 5 gas particles in 2 chambers:
Although there are 6 ways to distribute particles, assume 5 particles in one chamber represents ordered state. Therefore, 2 ways to represent ordered state and 4 ways to represent random distribution.

S = k(ln2)  (entropy for ordered state)
S = k(ln4)  (entropy for random distribution)
 

Now consider 1 mol gas particles in A and nothing in B:

Each particle has equal probability of being in either A orB.
Assume ordered state is for all particles to be in chamber A.
Since only one way to represent ordered state,
S = k lnW = k (ln 1) = 0   (entropy for ordered state)

Since 2^6E23 ways to represent random distribution,
S = k lnW = k (6x10²³ ln 2) = 5.7 J/K  (entropy for random distribution)
 

A gas expands spontaneously because the state of greater volume is more probable (higher S). When volume increases from V₁ to V₂,
DS = R ln(V₂/V₁)