10 independent observations are chosen at random from an exponential distribution with mean 1.
It asks that we calculate the probability that at least 5 of them are in the interval (1, 3)
I believe that I have to use the equation $(^{10}_5)p^5(1-p)^{10-5}$ but not exactly sure how to get p and how to check between the interval (1,3)
Use the definition of the exponential distribution to work out the probability that a single observation is in the interval $[1, 3]$. If you're not sure how to do that, then consider that $P(1 \leq X \leq 3) + P(X \leq 1) = P(X \leq 3)$ (i.e. the probability that $X$ is in that interval can be found through the difference of the probability that it is less than each of the endpoints).
Then, as you seem to have guessed, you take that probability and consider it to be the $p$ of a binomial distribution of the 10 observations, and work out the probability that at least (and be careful to note that it's at least, not exactly) 5 of them lie in that interval.