Let A = {1,2,3,4,5,6,7}.Then the number of functions from A to A such that f(f(f(x)) = x is

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In this video, I am discussing a very interesting problem on functions and permutation and combination which is an advanced level.

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the number of elements upon which the function acts non-identically can only be multiple of 3

non-identity action: none
there is only 1 such function, that is f(x)=x

non-identity action: 3
there are (⁷₃)2! such functions
(multiplied by 2 because such non-identity function can be envisaged as cyclic-permutation)

non-identity actions: 6
there are (⁷ ₆)(⁶ ₃)2!2!/(2!)

(divided by 2 since we partition 6 into 2 identical orbits)

Total=1+70+280 = 351

spiderjerusalem

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