Problem of the Day: 3/9/13

In music, a single beat consists of one quarter note, which can be subdivided into two eighth notes. Each eighth note can be further subdivided into two sixteenth notes. How many four-beat rhythms can be constructed from quarter notes, eighth notes, and sixteenth notes?

Solution to yesterday’s problem:

All you need to do in this problem is write the factorial out as its entire product, like this:

$14\times13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1$

Then split each factor into its prime factors:

$7\times2\times13\times3\times2\times2\times11\times...$

We don’t need to go any further because we already have the greatest prime factor we’re going to find here: 13.