[Math] Number of ways to insert 4 wires into 8-pin 8P8C Ethernet connector

This is an Ethernet crossover cable to connect two Ethernet ports:

Let's say you forgot correct pinout. Can you try manually all possible ways to connect it?

OK, you may remember that pins 1, 2, 3 and 6 are used. How many permutations of wires are possible?

#!/usr/bin/env python3

import itertools

wires=["left_1", "left_2", "left_3", "left_6"]

print ("right_1, right_2, right_3, right_6")
for x in itertools.permutations(wires):
    print (x)

This is 4!=24:

right_1, right_2, right_3, right_6
('left_1', 'left_2', 'left_3', 'left_6')
('left_1', 'left_2', 'left_6', 'left_3')
('left_1', 'left_3', 'left_2', 'left_6')
('left_1', 'left_3', 'left_6', 'left_2')
...
('left_6', 'left_2', 'left_1', 'left_3')
('left_6', 'left_2', 'left_3', 'left_1')
('left_6', 'left_3', 'left_1', 'left_2')
('left_6', 'left_3', 'left_2', 'left_1')

Now if you forgot these pins, you may ask, how many ways there are to insert 4 wires in 8P8C connector?

#!/usr/bin/env python3

import itertools

wires=[1, 2, 3, 4, 5, 6, 7, 8]

for x in itertools.combinations(wires, 4):
    print (x)

This is 8 choose 4 or C(8,4)=70.

(1, 2, 3, 4)
(1, 2, 3, 5)
(1, 2, 3, 6)
(1, 2, 3, 7)
...
(4, 5, 6, 8)
(4, 5, 7, 8)
(4, 6, 7, 8)
(5, 6, 7, 8)

Now how many ways there are to connect two 8P8C connectors using 4 wires?

#!/usr/bin/env python3

import itertools

wires=[1, 2, 3, 4, 5, 6, 7, 8]

for x in itertools.combinations(wires, 4):
    for y in itertools.permutations(x):
        for z in itertools.combinations(wires, 4):
            print (y, "<->", z)

It's 117600:

(1, 2, 3, 4) <-> (1, 2, 3, 4)
(1, 2, 3, 4) <-> (1, 2, 3, 5)
(1, 2, 3, 4) <-> (1, 2, 3, 6)
(1, 2, 3, 4) <-> (1, 2, 3, 7)
(1, 2, 3, 4) <-> (1, 2, 3, 8)
...
(8, 7, 6, 5) <-> (4, 5, 6, 7)
(8, 7, 6, 5) <-> (4, 5, 6, 8)
(8, 7, 6, 5) <-> (4, 5, 7, 8)
(8, 7, 6, 5) <-> (4, 6, 7, 8)
(8, 7, 6, 5) <-> (5, 6, 7, 8)

This coincides with Number of non-attacking placements of 4 rooks on an n X n board (for 8*8 board). (Coincidentally? Not sure.)

(the post first published at 20251209.)

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