Question

an airline is given permission to fly 4 new routes of its choice. the airline is considering 16 new routes: 5 in florida,7 in california and 4 routes in texas. if the airlines selectes 4 new routes at random from the 16 possibilities, determine the probability that 2 are in florida and 2 are in texas

Answer 1

I was really hoping this was a traveling salesman problem. No matter, we are looking at a combinobatrics problem:
Pr(2 in FL and 2 in TX) = Number of ways to pick 2 in FL and 2 in TX divided by number of ways to pick 4 out of 16.
\[\frac{\dbinom{5}{2}*\dbinom{4}{2}}{\dbinom{16}{4}}\]
\[\frac{\frac{5!}{3!2!}*\frac{4!}{2!2!}}{\frac{16!}{12!4!}}\]
\[\frac{\frac{5*4}{2}*\frac{4*3}{2}}{\frac{16*15*14*13}{4*3*2}}\]
\[\frac{10*6}{2*5*14*13}=\frac{6}{14*13}=\frac{3}{91}\]