Probability using a continuous random variable with distribution exponential

Question

A system uses a component whose duration in years is a continuous random variable with distribution exponential with a mean of 4 years. If 20 of these components are installed and work independently, determine the probability that after 6 years, at least 7 of them remain running.

I did the following:

$E(x)=4=\frac{1}{\lambda}$

$\implies \lambda=\frac{1}{4}$

$P(x>6)=1-e^{\frac{-6}{4}}=1-e^{\frac{-3}{2}}$

But I'm not convinced what I did

Answer 1

Error: $P_6=P(x\gt 6)=e^{-\frac{6}{4}}$. General solution for $A$= prob at least 7 are running is (binomial) $A=\sum_{k=7}^{20} \binom{20}{k} P_6^k(1-P_6)^{20-k}$.