Andy C. answered • 05/26/18

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N = 2^k where k is the number of pairs of (0,e)

D = N/2

The number of permutations is then n! / [(d!)(d!)]

This is the same formula as the permutations where there are 2 or more items that are the same.

For example, the first pair, k=1, the ordered pair is (o,e)

2^k = 2^1 = 2

D=2/2 = 1

2! / (1!)(1!) = 2

So there are two permutations: oe and eo

For k=2, 2^2 = 4.

D=4/2 = 2;

So there are 4!/[(2!)(2!)] = (4*3)/(2*1) = 6 permutations of ooee:

ooee

oeoe

eooe

eeoo

eoeo

oeeo

So for k=4 pairs of (o,e) namely the permutations of ooooeeee

there are

2^4 = 16 ---> D=16/2 = 8

(16!)/[(8!)(8!)] =

(16*15*14*13*12*11*10*9)/(8*7*6*5*4*3*2) = <---- 8! cancels

(15*14*13*12*11*10*9)/(7*6*5*4*3) = <--- 16 cancels 8x2

(14*13*12*11*10*9)/(7*6*4) <----- 15 cancels 5*3

(2*13*2*11*5*9) / 2 <--- 14 cancels 7, 12 cancels 6, 10 cancels 4 by a factor of 2

13*2*11*5*9 <---- one pair of 2's cancel out

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