<p>Mint-
From everything I have heard, gaining admittance on the flute is very, very competitive and in many programs they admit very few each year from what I have heard. </p>
<p>With the joint program with Columbia and Juilliard, you could potentially get a science degree, because it is not a traditional joint degree, you get your UG from Columbia while taking lessons as Juilliard, then you get your masters from Juilliard (there are descriptions of the program on here, that is just a nutshell).</p>
<p>Some music programs allow joint degrees, some, like Bard , require it.</p>