(Originais Teachy 2023) - Question Hard of Mathematics

A shooter is participating in a target shooting competition. He has a 70% probability of hitting the target in a single shot. However, the competition consists of 10 rounds of shots, and the shooter wants to calculate the probability of hitting exactly 7 shots. Considering that each round is independent and that the probability of hitting or missing the target does not change throughout the competition, calculate the binomial probability of the shooter hitting 7 shots in 10 rounds. Consider using the binomial distribution formula: P(X=k) = (n choose k) * (p^k) * ((1-p)^(n-k)), where n is the number of trials, k is the desired number of successes, p is the probability of success in a single trial, and (n choose k) is the binomial coefficient, which can be calculated as n! / (k! * (n-k)!). After the calculations, discuss how the probability of hitting 7 shots in 10 rounds compares to the probability of hitting 5 or 6 shots, and which factors influence these probabilities.