How many ways can 12 kindergartners line up to go to lunch?

Answer 1

The answer is 12! (said "twelve factorial"), which is equal to 12*11*10*9*8*7*6*5*4*3*2*1 or 479,001,600. Most people think about it like this:

There are 12 different people that can be first in line. After choosing the first person in line, there are 11 people left that can be second in line. After choosing the second person in line, there are 10 people left that can be third in line... and so on. The total number is the product 12*11*10*9*8*7*6*5*4*3*2*1 because, working in this way, each time a person is chosen to occupy "position n" in the lineup the next position can be filled in any of (12-n) different ways and we can only account for the total number of possibilities by multiplying the number of choices at presented at each stage in the process (if the last point is confusing, you might think about choosing a card from a deck of 52: you can select a suit--hearts, clubs, diamonds, spades--in one of four ways and you can select a rank--2, 3, 4, ... , queen, king, ace--in one of 13 ways. Thus the total number of possibilities in choosing a single card is 4*13=52).