Chance of exceeding 5 red balls out of 10, when picking from an infinite resevoir with a 30% red 70% blue.
Instead of "infinite number of balls" you better go for "very large number of balls".
This because it is rather weird to speak of e.g. "30% of infinite".
Further you are dealing with binomial distribution here.
By odds $30/70$ instead of $50/50$ you cannot just do it with "numbers of combinations" because the combinations are not equiprobable.
Apply binomial distribution with parameters $n=10$ and $p=30/100=0.3$.