Maximum likelihood estimation for exponential distribution

I would like to describe some gathered data for arrival process which seems not following Poisson distribution for people finding a job.
The data consists of two subpopulations (but we have only the combined results), first subpopulation where people assigned to the correct place (correct place means there is demand for their skills) and one where they are not. For both subpopulations, 23% will never find a job. After an initial period of (say) one year, people in subpopulation i who don't have a job yet and are part of the 77% will find a job at rate lambda_i. So we want to find (for example) the probability distribution of the number of people who find a job in one year for the population where the time in which an individual finds a job follows an exponential distribution with parameter lambda_1?
for example:
Let X_1, ..., X_10 ~ EXP(c) denote the time that someone finds a job in years. The time it takes for the first one to find a job is distributed as Y_1 ~ EXP(10c). Then the additional time for the second one to find a job is Y_2 ~ EXP(9c), etc.
for this, I will perform maximum likelihood to estimate the parameters.
I'm missing something here and can't see how to start my MLE.
I'll be grateful if someone would like to help.