To understand why the standard deviation of a binomial distribution with a mean of 25 lies in the interval [0, 5), we need to understand the properties of a binomial distribution and how it relates to the mean and standard deviation.

A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success. The mean of a binomial distribution is given by the formula np, where n is the number of trials and p is the probability of success in each trial.

In this case, the mean of the binomial distribution is 25. This means that np = 25. Let's assume that n = 100, which means we have 100 trials. So, p = 25/100 = 0.25.

The standard deviation of a binomial distribution is given by the formula sqrt(np(1-p)). Substituting the values, we get sqrt(100 * 0.25 * (1 - 0.25)) = sqrt(18.75) ≈ 4.33.

Now, let's consider the interval [0, 5). This means that the standard deviation can take any value greater than or equal to 0 but less than 5.

Since the standard deviation of the binomial distribution is approximately 4.33, which is less than 5, it lies within the interval [0, 5). Therefore, the correct answer is option 'A'.

In summary:
- The mean of a binomial distribution is given by np.
- The standard deviation of a binomial distribution is given by sqrt(np(1-p)).
- In this case, the mean is 25.
- Substituting the values, we find that the standard deviation is approximately 4.33.
- The standard deviation lies in the interval [0, 5), which is the correct answer.