as long as the failure rates of the individual components are independant, the time to first failure is distributed as exponential with mean 10.
You can show this rigorously by noting that the exponential distribution is the waiting time of a poisson process, and that the sum of iid poisson variables is also poisson with the parameters added up.
Let T be the time to failure of the System
P(T<k) = P(at least 1 component dead by k)
P(T<k) = 1-P(all components alive at k)
P(T<k) = 1-(P(component 1 alive at k)*P(component 2 alive at k).....)
You can easily work out the chance of each component being alive at K, since you were given the distribution for each component.
let T be the failure time of the system
P(T<k) = P(all components dead at time k)
P(T<k) = P(first component dead at time k) * P(second component dead at time k) * ...etc
Each term on the right hand side is the CDF of 1 component (which you know). Multiply them together to get your answer