SOLUTION: In how many ways can nine keys be arranged in a keychain

Question 1177518: In how many ways can nine keys be arranged in a keychain
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52886)   (Show Source): You can put this solution on YOUR website!
.

9 distinguishable items can be arranged in 9! = 9*8*7*6*5*4*3*2*1 = 362880 different ways.

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This problem is on PERMUTATIONS.

On Permutations, see introductory lessons
    - Introduction to Permutations
    - PROOF of the formula on the number of Permutations
    - Simple and simplest problems on permutations
    - Special type permutations problems
    - Problems on Permutations with restrictions
    - OVERVIEW of lessons on Permutations and Combinations
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Combinatorics: Combinations and permutations".

Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.

Answer by greenestamps(13209)   (Show Source): You can put this solution on YOUR website!

The problem is not well defined....

The other tutor answered the problem as though the keys were arranged in a row; that is certainly one interpretation.
9! = 362880

Another possibility is that the key chain is circular; then the problem is like arranging 9 people around a table. The number of arrangements in that case is

9!/9 = 8! = 40320

Additionally, most keys have a flat side and a jagged side, so each key could be on the chain in either of two ways. That would make the number of arrangements

(8!)(2^9) = 20643840

Or if the key chain is not circular and each key can be on the chain in either of two ways, the total number of arrangements is

(9!)(2^9) = 185794560

If this is actually a problem that you are supposed to solve, ask the person who asked you the question what the groundrules are....

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