(Combinatorics)  A bicycle manufacturer decided to offer electric bicycles to university students. 

35 students got a red colored bicycle and another 10 students got a green colored bicycle. Another 15 students changed their mind and decided not to accept
the offer. In how many ways can the bikes be allocated?
I understand that this should be (60 choose 35) for the red bikes and (25 choose 10) the green bikes. but what I don't understand is the part about the students who changed their mind. should I choose them from the remaining students so (15 choose 15) or should they be chosen randomly from all the students (60 choose 15). any help would be appreciated!
Discrete Mathematics
Zero1 Zero1
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1 Answer

You are right. There is a toal of 60 students. You first choose 35 students who get a red bicycle, then choose 10 students who get a green bicycle, and then the remaining 15 students will get no bicycle in ${15\choose 15}=1$ ways. So the number of different ways is going to be
$${60\choose 35}{25\choose 10}{15\choose 15}.$$

You can do this in any order, the number of different ways is also equal to 
$${60\choose 10}{50\choose 35}{15\choose 15}.$$
This means you first chose the 10 students who get no bicycle, the choose students who get a red bicycle, and then the ones who get a green bicycle. 
Or 
$${60\choose 15}{45\choose 35}{10\choose 10}.$$

You can check that all this formulas give you the same number.
Daniel90 Daniel90
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