A system is subjected to shocks of type 1, 2 and 3, which are generated by independent homogeneous Poisson processes with respective intensities $\lambda_1=0.2$, $\lambda_2=0.3$ and $\lambda_3=0.4$. A type 1 shock causes a system failure with with probability 1, a type 2 shock causes a system failure with probability 0.4 and a shock of type 3 causes a system failure with probability 0.2. The shocks occur permanently whether the system is operating or not.
Three shocks occur in the interval[0, 10 hours]. What is the probability that the system does not experience a failure in this interval?
My attempted solution is as follows
Let A be the event that the system survives
P(A)=P(No type 1 shocks occur in the interval)*P(Three non fatal shocks of type 2 or 3 occur in the interval)
P(No type 1 shock occur in [0,10])=$e^{−0.2∗10}=e^{−2}$
I would like help on how to get the other probability i.e. P(Three non fatal shocks of type 2 and type 3 occur in the interval)