math question

Question

A party of 27 waited at the bus stop for a double decker bus which can carry 15 in the
lower deck and 12 in the uppder deck, how many ways the people can be seated if there
are 5 who want to be in the lower deck and 6 who stictly want to be in the upper deck?

Answer 1

Assuming that the persons who have a strong preference get their way, that puts 5 in lower deck & 6 in upper deck, as they wanted.  That leaves 16 people to sort themselves out by pushing and shoving (and perhaps going up the stairs).

 

The fulfillment of the upper deck can be done in 16C6 ways. That's 8008.

The rest of the people take the seats in the lower deck, period.

 

So the OVERALL ANSWER IS 8,008. You get the same answer if you find # of ways of putting 10 of the 16 people in the lower deck, instead!