Octie!!

Probability Level 4

Octie the octopus has one shoe and one sock for each of it's eight legs.How many ways can he put on the shoes and socks provided he must put on a sock before a shoe on each leg?


The answer is 81729648000.

Eddie The Head by Eddie The Head , 27, India

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1 solution

Eddie The Head
Mar 6, 2014

If there wasn't any restriction then he could put on the 16 items in 16 ! 16! ways. Let us label each of it's 8 legs as L 1 , L 2 , . . . . . , L 8 L_1,L_2,.....,L_8 . First half of these 16 ! 16! cases he will put the sock before the shoe in leg L 1 L_1 and hence number of valid combinations wrt L 1 L_1 is 16 ! / 2 16!/2 Similarly if we consider the leg L 2 L_2 ,then the number of ways = 16 ! / 2 2 16!/2^{2} Continuing in this way pto the 8th leg we eill have 16 ! / 2 8 16!/2^{8} valid cases....Hence the answer is 16 ! / 2 8 = 81729648000 16!/2^{8} = 81729648000

16 Helpful 0 Interesting 0 Brilliant 0 Confused

I thought of it like when you are asked how many distinct ways there are to organize the letters in banana: 6 ! / ( 3 ! ∗ 2 ! ) 6!/(3!*2!) . Instead of considering the same letters to be equal, here we consider garments going on the same leg to be equal in a sense, so we end up diving by 2 ! ⋯ 2 ! = 2 8 2!\cdots2!=2^8 . Of course, this analogy isn't quite right. For the words, we divide by the factorial because all permutations of the same letter really are the same. Here, we divide by 2 2 because only one of the two possible permutations of each leg's garments is permitted.

Nick Stanford - 7 years, 1 month ago

thnkx..nice solution

Max B - 7 years, 1 month ago

one question...what's the gender of octopus...in ur solution...!!!!..just kidding..you have mentioned it once as he and then as she...but dont mind

Max B - 7 years, 1 month ago

very, very long factorial number (y)

Dani Natanael - 7 years ago

Why is the answer not 8 ! * 8 ! ? Can't we just just arrange the socks and then arrange the shoes ?

Deepak Kandasamy - 7 years, 1 month ago

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Beauae that also includes putting shoe first...

Ayan Bhuyan - 7 years, 1 month ago

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