When should I use the *Central Limit Theorem*?

Question

I am facing the following question:

Assume you have invited a $100$ people to a party. The probability that one would decide to come to the party is $0.75$.

What is the probability that more 70 but not more than 80 people will decide to come to your party?

I can tell that if $X$ is the number of people that have decided to come to the party, I can easily say that $X \sim Bin(100, 0.75)$.

I can use that to tell what is $P(70 < k=x <80)$

Why would I want to use the Central Limit Theorem and move to $N \dot{\sim}(100\cdot0.75, 100\cdot0.25)$?

Answer 1

If you have a good calculator that will be able to handle binomial coefficients with $n= 100$, it's feasible, if a bit tedious, to compute and add $10$ probabilities. But what if you invited $10$ million people and wanted to know the probability that between $7,490,000$ and $7,500,000$ would come?

(I guess if you can afford to throw that big a party, you can afford to hire someone to compute that for you)