To find the value of p in a binomial distribution, we need to use the formula for the mean and standard deviation of a binomial distribution.

The mean of a binomial distribution is given by the formula: mean = n * p, where n is the number of trials and p is the probability of success in each trial.

The standard deviation of a binomial distribution is given by the formula: standard deviation = sqrt(n * p * (1 - p)), where n is the number of trials and p is the probability of success in each trial.

Given that the mean is 20 and the standard deviation is 4, we can set up two equations to solve for p.

1. mean = n * p
20 = n * p

2. standard deviation = sqrt(n * p * (1 - p))
4 = sqrt(n * p * (1 - p))

Now, we can solve these equations simultaneously to find the value of p.

From equation 1, we can rearrange it to solve for n:
n = 20 / p

Substituting this value of n into equation 2, we get:
4 = sqrt((20 / p) * p * (1 - p))

Simplifying the equation, we have:
4 = sqrt(20 * (1 - p))

Squaring both sides of the equation, we get:
16 = 20 * (1 - p)

Dividing both sides of the equation by 20, we have:
16 / 20 = 1 - p

Simplifying, we get:
0.8 = 1 - p

Subtracting 1 from both sides of the equation, we get:
-0.2 = -p

Multiplying both sides of the equation by -1, we have:
0.2 = p

Therefore, the value of p is equal to 0.2, which is equivalent to 1/5. So, the correct answer is option 'C', which is 1/5.