= Solution
Partition the enumeration of $B$ into the pairs $\{b_{2j-1},b_{2j}\}$, for $j\geq1$. An infinite <binary sequence> $x$ determines a <bijection> $h_x:B\to B$: leave the $j$th pair fixed when $x_j=0$, and interchange its two elements when $x_j=1$. Each element stays in its own pair, so this function is a <bijection>, indeed its own inverse.
Different <binary sequences> give different <bijections>, since a changed bit changes the action on the corresponding pair. We have therefore embedded an <uncountable set> into $S_3$. Hence \b[$S_3$ is uncountable], proving <uncountability of permutations of a countably infinite set>.
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