Answer
$220$ different ways.
Work Step by Step
1) It is not a sequence of choices.
2) The order in which the colleges are selected does not matter.
It is a combination of 12 distinct colleges taken 3 at a time:
$_{12}C_3=\frac{12!}{3!(12-3)!}=\frac{12!}{3!\times9!}$
But, $12!=12\times11\times10\times(9\times8\times7\times6\times5\times4\times3\times2\times1)=12\times11\times10\times9!$
$_{12}C_3=\frac{12\times11\times10\times9!}{3!\times9!}=\frac{12\times11\times10}{3\times2\times1}=\frac{1320}{6}=220$