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I'm currently working on a assigment, where I have to find out the number of possible combinations when coupled 5 wires of the plugboard and Generalize the calculation to reflect the number of possible combinations of "n" wires.

I did find out how many combinations the 5 wires make which is

(26*25)/2*(24*23)/2*(22*21)/2*(20*19)/2*(18*17)/2*1/5! = 5.019.589.575

Now I need to found out how I make a formela for "n" wires. Can anybody help?

Skooou
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Note that your numerator is $\frac {26!}{16!}$ Can you replace the $16$ with an expression that depends on $n$? Think about the $n=1$ and $n=2$ cases. Then the $5!$ becomes ... and how many factors of $2$ are in the denominator?

Ross Millikan
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  • I'm not a 100% sure I understand what you mean. What do you mean that the numerator is 26!/16!. – Skooou Dec 16 '15 at 17:45
  • You are multiplying all the numbers from $26$ down to $17$. One way to express that is to multiply all the numbers from $26$ down to $1$ which is $26!$ and divide by all the numbers from $16$ down to $1$, which is $16!$. This shows up in many combinatorial problems. – Ross Millikan Dec 16 '15 at 17:55
  • So it would be something like (26!/16!)/32*1/5!=5019589575 right? Now I need to replace the 16! with an expression that depends on n, but how do I do that? – Skooou Dec 16 '15 at 17:59
  • Well 16! can be (26-10) or (26-25) - 5 is wires which have 2 endes and therefor we med 2. (26!/(26-25)!)/321/5! – Skooou Dec 16 '15 at 18:03
  • and is there a way to move the /32 or do something with this? – Skooou Dec 16 '15 at 18:08
  • You divide by $2$ once per wire because you can choose the ends in either order. That would become a factor $2^n$ in the denominator. You divided by $5!$ because you can change the order of the wires that many ways. What would that be for general $n$? – Ross Millikan Dec 16 '15 at 18:39
  • It would be $26!/(26-2n)!1/2^n$, but it doesn't give the right answer. EDIT it would be $26!/((26-2n)!2^n )1/n!$ right? – Skooou Dec 16 '15 at 18:47
  • Yes, that is correct – Ross Millikan Dec 16 '15 at 19:06