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Serial Number Euro Truck Simulator 2 Version 1.3.1

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Probability Question using the Inclusion-Exclusion principle.

I’m not quite sure if I’m using the Inclusion-Exclusion principle correctly, and would appreciate any help.
QUESTION:
There are 50 prisoners, 4 of which are all the subject of the “A” classification, and the rest are all the subject of the “B” classification. A guard starts to choose prisoners at random, and gives a maximum of 2 of the “A” category, 4 of the “B” category. What is the probability that the guard chooses a “B” prisoner?
PS: the additional information I mentioned is my attempt at a solution. However, I am still unsure if I’m using the principle correctly.
MY ATTEMPT
For the total number of outcomes = $ 4cdot 50 = 200$ (I believe)
For the number of outcomes $ = binom{200}{0} + binom{200}{1} + binom{200}{2}$
$ = binom{200}{0} + binom{200}{1} + binom{200}{2} – binom{200}{0,1} – binom{200}{0,2} – binom{200}{1,2}$
$= 3^5 – 5^5$
$= 5625 – 25$
$= 301$
MY CONCERNS
Am I using the principle correctly? (see example)
Is there an easier way of answering this question?

A:

You forgot the zero category of the classification. So we should write $binom{200}{1} – binom{200}{0,1}$, for example.
You also underestimated the number of events, as $binom{200}{1}$ is not the right way to write $binom{200}{1}$.
My answer is more or less random but close. I chose to start with the good prisoners and count the number of ways you

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