ways of arranging double decker bus passengers

[galactus]

I am always interested in a new counting problem.

What is the way to go about this one?.

A double decked bus can accomodate 110 passengers,50 in the upper & 60 in lower in how many ways can?
passengers be accomodated if 15 refuse to be in upper deck while 10 refuse to be in lower deck.

I was thinking, since 15 refuse to be on the upper, then they must be on the lower deck. That makes 75. Minus the 10 who refuse to be on the lower and then must be in the upper equals 65. That puts 45 in the upper deck. That equals 110, but
it's rather confusing.

 

[pka]

A double-decked bus can accommodate 110 passengers, 50 in the upper & 60 in lower. In how many ways can passengers be accommodated if 15 refuse to be in upper deck while 10 refuse to be in lower deck?

As good capitalist we go ahead and seat the people who have a preference accordingly. This leaves 85 passengers that can be seated on either deck. We can choose 40 to be placed on the upper deck and the others are seated on the lower deck.
So the answer is \(\displaystyle {85} \choose {40}\).

 

[galactus]

Wow, pka, I thought it'd be more complicated than that. I see.

That's a huge number.