Why is n the number of possible ways to arrange n distinct items?

[zeion]

Homework Statement

Why?

Homework Equations

The Attempt at a Solution

I know that n! = n(n-1)...1
and if n = 3 then the possibility for the first entry is 1 or 2 or 3, and if the first entry is 1 then the second cannot be 1 so the second entry has 2 possibilities. But why do I multiply

 

[symbolipoint]

Because n! is literally a generalization from direct human experience. Try this yourself! See how many ways you can fill n slots using n distinct objects; start small, very small, and continue upward. Try with poker chips, marbles, ... anything you want.

 

[gb7nash]

Think of there being n slots. You want to find all possible arrangements.

How many ways of choosing an item for the first slot? We have n items to choose from, so it must be n.

Once we placed an item in the first slot, how many ways are there of choosing an item (from the remaining items) for the second slot?

After we placed the first two items, how many ways of choosing an item for the third slot?

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Once you have them all, you should know the multiplication rule and...
(If you're not sure what the multiplication rule is, look up the definition and think about how it relates to the problem)

 

[vela]

Say you have 4 objects, A, B, C, and D, and you want to fill two slots. You have these possibilities.

AB AC AD
BA BC BD
CA CB CD
DA DB DC

Each row corresponds to a choice of filling the first slot, and the entries across each row just go through the possibilities of filling the second slot. So you can see the total number of possibilities is just the number of rows times the number of columns. Do you see how it generalizes?