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Q)

$S=\{1,2,3..............12\}$ $A,B,C $ are disjoint equivalent subsets of $S$ so that $A\cup B\cup C=S$. How many such sets $A,B,C$ can be formed?

$\begin{array}{1 1} (A) ^{12}C_4\times^8C_4\times^4C_4 \\ (B) ^{12}C_4\times^{12}C_4\times^{12}C_4 \\ (C) ^{12}P_4\times^8P_4\times^4P_4 \\ (D) 64 \end{array} $

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A)
Since the sets $A,B,C$ are disjoint and equivalent subsets of S,
each set has four elements.
Set A can be formed in $^{12}C_4$ ways
Set B can be formed in $^8C_4$ ways and
Set C can be formed in $^4C_4$ ways.
$\therefore$ The reqired no. of sets = $^{12}C_4\times^8C_4\times^4C_4$
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