Without doing any computation, put the following in order from least to greatest, assuming the population is normally distributed with $\mu=100$ and $\sigma=15$.
(a) $P(90 \leq \bar{x} \leq 110)$ for a random sample of size $n=50$
(b) $P(90 \leq \bar{x} \leq 110)$ for a random sample of size $n=40$
(c) $P(90 \leq x \leq 110)$

Step 1 ：The problem is asking us to compare the probabilities of different events in a normally distributed population. The events are the probabilities that the sample mean falls within a certain range, for different sample sizes, and the probability that a single observation falls within that range.

Step 2 ：The key to solving this problem is understanding the concept of the standard error of the mean. The standard error of the mean is a measure of the dispersion of the sample mean around the population mean. It is equal to the standard deviation divided by the square root of the sample size.

Step 3 ：As the sample size increases, the standard error decreases, meaning that the sample mean is more likely to be close to the population mean.

Step 4 ：Therefore, for a larger sample size, the probability that the sample mean falls within a certain range around the population mean is higher.

Step 5 ：So, the order from least to greatest should be (c), (b), (a).

Step 6 ：\(\boxed{\text{The order from least to greatest is (c), (b), (a).}}\)