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

  • Thread starter zeion
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In summary, the conversation discusses the concept of n!, which represents the number of ways to fill n slots with n distinct objects. This is a generalization based on direct human experience. The conversation includes an example of using poker chips to demonstrate the concept and explains the use of the multiplication rule to calculate the total number of possibilities.
  • #1
zeion
466
1

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
 
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  • #2
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.
 
  • #3
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?

.

.

.

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)
 
  • #4
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?
 

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

This is because when arranging distinct items, the first item can be placed in n different positions, the second item can be placed in n-1 positions (since one position is already taken by the first item), the third item can be placed in n-2 positions, and so on. This results in n*(n-1)*(n-2)*...*3*2*1 ways to arrange n items, which is equal to n! (n factorial).

2. What does it mean for items to be "distinct" in this context?

In this context, distinct items means that each item is unique and cannot be exchanged for another item. This means that if the items were rearranged, it would result in a different arrangement.

3. Can the number of possible arrangements for n distinct items be greater than n! ?

No, the number of possible arrangements for n distinct items will always be n! or less. This is because n! is the maximum number of ways to arrange n items, and if any items are repeated or not distinct, it will result in a smaller number of possible arrangements.

4. Is the number of possible arrangements affected by the order in which the items are arranged?

Yes, the number of possible arrangements will be affected by the order in which the items are arranged. This is because changing the order of the items will result in a different arrangement, even if the items are the same.

5. What is the significance of the number of possible arrangements for n distinct items?

The number of possible arrangements for n distinct items has many practical applications in mathematics, statistics, and computer science. It is used to calculate probabilities, permutations, and combinations, and is also used in algorithms and data structures. It is an important concept in understanding the complexity and efficiency of various processes and systems.

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