(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!

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. 
$${60\choose 15}{45\choose 35}{10\choose 10}.$$

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

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