Final answer:
The investor's profit from selling the European put option would be the premium received, which is $5.81, since the option will not be exercised by the holder.
Step-by-step explanation:
When an investor sells a European put option, they receive a premium and have the obligation to buy the underlying asset at the strike price if the option is exercised. In this scenario, the investor sold the put option for a premium of $5.81 with a strike price of $75. If in one month the stock price is at $77.34, the option would not be exercised by the holder since the market price is higher than the strike price, meaning the investor gets to keep the premium as profit. Therefore, the profit for the investor would be the premium received, which is $5.81.