$A = \left\{ {{a_1},{a_2},{a_3},{a_4}} \right\}$
$B = \left\{ {{b_1},{b_2},{b_3}} \right\}$
${A_j} = \left\{ {f:\left. {A \to B} \right|f({a_i}) \ne {b_j},1 \leqslant i \leqslant 4} \right\},1 \leqslant j \leqslant 3$
$\left| S \right| = {3^4} = 81,\left| {{A_j}} \right| = {2^4} = 16$
$\left| {{A_1} \cap {A_2}} \right| = \left| {{A_1} \cap {A_3}} \right| = \left| {{A_3} \cap {A_2}} \right| = 1$
$\left| {{A_1} \cap {A_2} \cap {A_2}} \right| = 0$
$\left| {{A_1} \cap {A_2} \cap {A_3}} \right| = 81 - (3 \times 16 - 3 \times 1 + 0) = 36$