What Creature Has 1 Head, 6 Arms And 2 legs?

Probability Level 2

The above image is a common dance pose, used to create the illusion of a multi-limbed deity. The dancer in the front has her hands in the down position, the dancer in the middle has her hands in the center position and the dancer in the back has her hands in the up position.

Other than keeping hands to the respective sides, there is no restriction on the positions. For example, the dancer in the middle could have her left hand in the down position, and her right hand in the up position.

How many ways can 3 dancers recreate this dance illusion?

Image credit: Wikipedia

If you want another problem on multi-limbs, check out The Math Behind The Perfect Man .

3 3 ! 3! ( 3 ! ) 2 (3!) ^2 ( 3 ! ) ! (3!)!

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Arron Kau by Arron Kau

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8 solutions

Prasun Biswas
Mar 13, 2014

It is said that the 3 dancers can change the position of their hands only on the side that the hand points to, i.e., the left hand of one cannot make a position of right side and vice-versa.

Then, the hands that recreate the left part of the dance pose can be rearranged in 3 P 3 = 3 ! 3P3=3! ways as there are 3 3 hands on left side (one from each person) and the other 3 3 hands of the dancers recreate the right side of the dance pose can also similarly be arranged in 3 P 3 = 3 ! 3P3=3! ways.

So, total no. of ways = 3 ! × 3 ! = ( 3 ! ) 2 =3!\times 3! = \boxed{(3!)^2}

29 Helpful 0 Interesting 0 Brilliant 0 Confused

what does '!'sign after 3 mean ??

Rishabh Tripathi - 7 years, 2 months ago

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The '!' sign refers to factorial of a number. For example, n ! n! is pronounced as "n factorial" and it is calculated as the product of all natural nos. till n n , i.e.,

n ! = n ( n − 1 ) ( n − 2 ) . . . . . 3.2.1 = 1.2.3.4..... ( n − 1 ) n n!=n(n-1)(n-2).....3.2.1 = 1.2.3.4.....(n-1)n

You can write it either way...

Prasun Biswas - 7 years, 2 months ago

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thanks for the answer, I have now understood the meaning of ! in the answers

Patricia Ann Buiser - 7 years, 2 months ago

Factor is denoted as '!'

Shrikrishna Oza - 7 years, 2 months ago

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Not factor, n ! n! denotes factorial of a number n n . There is a lot of difference between factor and factorial...

Prasun Biswas - 7 years, 2 months ago

factorial

Ali Asghar Qaiyum - 7 years, 2 months ago

i don't understand what ! mean

Subrota Roy - 7 years, 2 months ago

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@Subrota Roy Hmm....see my reply to the first comment made by Rishabh Tripathi in this discussion. He had the same doubt and I clarified it in my reply. If you have any further doubts, drop me a line in the comments... :)

Prasun Biswas - 7 years, 2 months ago

Can you prove beal conjecture

Avinash Jha - 7 years, 2 months ago

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I don't see how it is related to this problem !! Also, I don't even know what Beal's conjecture is, let alone prove it.

Prasun Biswas - 7 years, 2 months ago

27

Mohamed Larbi Chtaibi - 7 years, 2 months ago

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27..what ?? The answer is ( 3 ! ) 2 (3!)^2 which is equal to 36 36 .

( 3 ! ) 2 = ( 3 × 2 × 1 ) 2 = 6 2 = 36 (3!)^2=(3\times 2\times 1)^2=6^2=\boxed{36}

I think it would be good for me and also others if comments are made such that others can understand what the person is trying to say!!

Prasun Biswas - 7 years, 2 months ago
Arijit Banerjee
Mar 14, 2014

No. of ways of changing positions of hands is 3! and no. of ways they can change their positions is 3! ... so total number of ways they can perform this dance is (3!)^2

3 Helpful 0 Interesting 0 Brilliant 0 Confused

what is the meaning of exclamation point in the problem.question ?

Reginald Pilar - 7 years, 2 months ago

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" ! " means factorial..3! = 3 * 2 * 1 = 6

Arijit Banerjee - 7 years, 2 months ago
Raj Saha
Mar 27, 2014

Total position =3, And each people has 2 hands. So the arrangement of hands without restriction one side is 3!. same for both sides simultaneously of hands is 3!^2

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Srijan Kumar
Mar 26, 2014

Solved it in first attempt. https://www.youtube.com/watch?v=FJbmB9k2Y88

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Danilo Nebit
Mar 25, 2014

In each side, there are three positions and three dancers to fill each of the positions. For example, on the right side, there are three right hands (from the three dancers) that can fill the top most position. After the topmost, there would be two right hands then the same with the bottom-most. So, on the right side there are 3! possible combinations. The same with the left side. So we have (3!)^2

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Henrique Ribeiro
Mar 24, 2014

N possibilities 3 x K possibilities 3 = 9 or 3²

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Julieta Ramos
Mar 20, 2014

3 positions for each limb of 3 girls = (3)^2

0 Helpful 0 Interesting 0 Brilliant 0 Confused

I think there are a few more possibilities.

Josh Silverman Staff - 7 years, 2 months ago

u missed the factorial sign

Arijit Banerjee - 7 years, 2 months ago
Rakesh Yadav
Mar 20, 2014

possible way for arrange the position of three dancer 3!,possible way for arrange there hands is also 3! so total way is (3!)^2

0 Helpful 0 Interesting 0 Brilliant 0 Confused

total is (3!)

Ridzmar Narawi - 7 years, 2 months ago

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