The set S = {1, 2, 3, . . . , 12} is to be partitioned into three sets A, B and C of equal size. Thus,A∪B∪C=S,A∩B=B∩C=A∩C=ϕThe number of ways to partition S is

# The set S = {1, 2, 3, . . . , 12} is to be partitioned into three sets A, B and C of equal size. Thus,$\mathrm{A}\cup \mathrm{B}\cup \mathrm{C}=\mathrm{S},\mathrm{A}\cap \mathrm{B}=\mathrm{B}\cap \mathrm{C}=\mathrm{A}\cap \mathrm{C}=\mathrm{\varphi }$The number of ways to partition S is

1. A

$12!/3!\left(4!{\right)}^{3}$

2. B

$12!/3!\left(3!{\right)}^{4}$

3. C

$12!/\left(4!{\right)}^{3}$

4. D

$12!/\left(3!{\right)}^{4}$

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### Solution:

.Required number of ways

$\begin{array}{c}{=}^{12}{\mathrm{C}}_{4}{×}^{8}{\mathrm{C}}_{4}{×}^{4}{\mathrm{C}}_{4}\\ =\frac{12!}{8!4!}×\frac{8!}{4!4!}×1=\frac{12!}{\left(4!{\right)}^{3}}\end{array}$

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