At a local pizza parlor, patrons have 3 choices of cheese and 5 choices of meat. In how many different ways can a patron choose 1 type of cheese and 1 type of meat?

##15##

##15## is correct. In general, the number of ways to choose ##k## objects from a set of ##n## objects is ##((n),(k)) = (n!)/(k!(n-k)!)##

When choosing one object, this simplifies nicely. ##((n),(1)) = (n!)/(1!(n-1)!) = (n!)/((n-1)!) = n##

(note that this is intuitively obvious, as there are clearly only ##n## choices when choosing a single object from ##n## objects, however the above formula is useful for more complex problems... for example, selecting three types of meat and two types of cheese)

As the choice of meat and the choice of cheese are independent of each other, we can simply multiply them to get the total number of choices. Again, the intuition is that for each type of meat, we add ##"cheese choices"## to the total number of choices, meaning we can just multiply ##"cheese choices"## by ##"meat choices"##

Thus, we have

##"Total choices" = "meat choices" xx "cheese choices"##

##=((5),(1))xx((3),(1))##

##=5xx3##

##=15##