Probability with equally likely outcomes needs counts, and listing every outcome (as with trees and tables) stops being practical quickly. Counting rules do the listing for you. The first is for arrangements, where order matters.
Key idea
Multiplication principle: if a task has steps with $n_1, n_2, \ldots$ choices, there are $n_1 \cdot n_2 \cdots$ ways to do it. Arranging $n$ different objects in a row: $n! = n(n-1)\cdots 2 \cdot 1$ ways. Arranging $r$ of $n$ different objects in order:
$$_{n}P_{r} = n(n-1)\cdots(n - r + 1) = \dfrac{n!}{(n-r)!}.$$
Why it works
Fill the positions one at a time. The first position can hold any of the $n$ objects. Whichever you chose, $n - 1$ remain for the second position, and so on. Every first choice combines with every second choice, which is why the counts multiply; a tree diagram shows the branches multiplying. For $_nP_r$ you stop after $r$ positions, which is $n!$ with the unused tail $(n - r)!$ divided away. (By convention $0! = 1$, so $_nP_n = n!$.)
Worked example: A full arrangement
In how many orders can 5 different books stand on a shelf?
- 5 choices for the first spot, 4 for the second, 3, 2, then 1.
- $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$.
Answer: 120
Worked example: Some of the objects, in order
A club of 10 elects a president, a vice president and a secretary (different people). How many outcomes are possible?
- Order matters: being president is different from being secretary.
- $_{10}P_{3} = 10 \cdot 9 \cdot 8 = 720$.
Answer: 720
Worked example: Repeated letters
How many different arrangements of the letters in LEVEL are there?
- If all 5 letters were different: $5! = 120$.
- But the two Ls can swap without changing the word, and so can the two Es: each word was counted $2! \cdot 2! = 4$ times.
- $\dfrac{120}{4} = 30$.
Answer: 30
Common mistake
Using a permutation when order does not matter. Choosing 3 people for a committee is not like electing three officers: the committee {Ann, Ben, Cal} is one group however you list it, but $_{10}P_3$ counts it 6 times. Check: if you swap two of the chosen, is it a different outcome? If not, see combinations.