Question

There are 12 different pizza toppings, and Arturo must rank his top 6 in order. How many different possible rankings are there?

Answer 1

The first ranking can take 12  ways.
The second ranking can take 12 - 1 = 11 ways
The third ranking can take 12 - 2 = 10 ways
and so until the six rankings are performed.
The product of the six rankings is
12*11*10*9*8*7 = 665,280.

This result is given by the permutations of 6 objects out of 12. It is given by
$_{12}P_{6} \\ = \frac{12!}{6!} \\ = \frac{12*11*10*9*8*7*6!}{6!} \\ =12*11*10*9*8*7 \\ =665,280$
which yields the same result.

Answer: 665,280