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Chef and Strange Addition

Chef likes to solve difficult tasks. This time, he tried to solve the Big Famous Unsolvable $A+B=C$. One of his friends played a prank on Chef and randomly shuffled the bits in $A$ and $B$ (independently in each number). However, the funny thing is that the sum of the resulting numbers remained $C$ even after shuffling. Chef is now wondering: in how many ways is it possible to shuffle the bits of

HINT LADDERno hints yet
L1 Observation
L2 Technique
L3 Approach
L4 Pseudo-code
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L5 Full solution
L5 unlocks only if you insist twice
solution.cppC++17

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