Why this probability was calculated using Binomial Distribution?

I think the question is assuming that each individual woman has an 82% chance of getting married, independently of what other women will do.

We aren't choosing a 2- or 3-woman sample, we are merely checking the marital status of 20 women and checking if there happen to be 2 or 3 who are unmarried.

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EDIT: Another way of looking at the problem:

Let's say we have a ball pit filled with 1 million balls. 820,000 are blue and 180,000 are red. Therefore, if I pick a ball at random, I have an 82% chance of it being blue and a 18% chance of it being red.

Now, what if draw a blue ball, throw that ball away, and decide I want to draw another one? It's true that the probability distribution has changed, since there are now 819,999 blue balls and 180,000 red balls, with 999,999 total balls. But for simplicity's sake, we can assume the probability distribution it hasn't changed very much (only by ~$10^{-6}$ in fact), so keeping our 82%/18% distribution is still going to be mostly accurate.

If I draw a small number of samples relative to the total number of balls (~20 samples relative to 1 million), the distribution is approximately binomial.

So on a mathematical level, you are correct: the distribution does change when you sample without replacement, but I think the problem wants you to make a simplifying assumption.

The question is implying that there are enough 25 year old women (and a large enough sample population) to assume that the probability for any given woman is 0.82 and hence the trials are independent.

The probability of each woman being married in their lifetime is an independent Bernouilli trial (or at least assumed to be for the sake of the question), and consequently as this experiment for a given sample forms a sequence of independent Bernouilli trials, the Binomial distribution is quite suitable to use.