If there are 9 people, how many ways can 3 of them be lined up for a photo?

There are three slots: _ _ _ because we have three decisions to make: who will be the first person, who will be the second person, who will be the third person. In each slot, fill in the number of ways you can make each choice and then multiply them all together. There are 9 ways to choose the first person, 8 ways to choose the 2nd, and 7 ways to choose the third. In this case: 9*8*7=504.

As this question implies, the slot method counts each different ordering separately. That is, it is the same as using nPr for a problem like this. if you use it for something where order doesn't matter, make sure you divide out the repetitions.

it's also applicable to problems besides classic permutation problems. For example, if a class has 25 boys and 14 girls, how many ways can we choose one boy and one girl to do whatever?

Two slots: _ _

25 ways to choose the boy and 14 ways to choose the girl: 25*14=350. In problems like this where each selection is made from a different pool of people, you don't have to worry about unwanted repetitions, so you're done.