How do people make money from selling options contracts?


So ive been reading a bunch on this trying to figure out how options trading works. Things such as call or put options.

I feel like im starting to get some things and would like confirmation if im understanding it right. But I also have a follow up question. Possibly explain in the simplest to understand terms.

So say i have $100 dollars.
I can buy 1 share of some stock thats currently worth $100 and down the line if it goes up to $150, sell it for a $50 profit.

But with a call option. For example I can buy a contract for 1$ to have the right to buy 100 shares(which if i understand correctly is the quantity each contract is worth) of that same stock at $125 per share within the next month lets say. So I would have to spend $100 bucks still to buy 100 contracts a dollar each is that correct? It wouldnt be just a dollar for the whole 100 shares, right?
So assuming i paid $100 for this contract. If the price once again goes up to $150, i can buy 100 shares for $125 instead for a total of $12,500 and then sell them for a total of $15,000, netting myself a profit of $2,500 minus the $100 premium i paid and walk away with $2,400. Did i understand all that correctly?

My follow up question to that then stems from seeing that people dont actually exercise their option and instead sell it to someone else at a higher premium. This is what im having trouble understanding. Why would anyone else buy this option for a higher price from you, when the much higher premium they pay will probably cancel out any profit they would have made on the sale of that stock themselves. Can someone explain to me, how do those people make money from buying your call option contract that is now much more expensive for them, but only profitable to you since you paid barely any money for it?


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