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Three $N$ bit binary strings $S_1$,$S_2$,$S_3$ are selected in random. What is the probability that result of bit-wise XOR among them contains $k$ $1$'s.i.e. $S_1\oplus S_2\oplus S_3$ = $S$ , No of set bits in $S$ = $k$

 

is it $\binom{n}{k}\left ( \frac{1}{2} \right )^k\left ( \frac{1}{2} \right )^{n-k}$ ??
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I think your formula is valid bcoz we want odd number of 1's, hence (1,1,1), (0,0,1),(0,1,0),(1,0,0) are the possibilites making 4/8 as succes probability or 1/2. So for k bits to be set nC(1/2)k(1/2)n-k

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