JEE Main · 2007 · 29 Apr 2007 · Offline · Permutations and Combinations

The set

S = {1, 2, 3, \dots, 12}

is to be partitioned into three sets

A, B, C

of equal size. Thus,

A \cup B \cup C = S, A \cap B = B \cap C = A \cap C = ϕ

. The number of ways to partition

S

Answer: (C)

Solution

Since the three groups are named A, B and C, they are treated as distinguishable, each to receive four of the twelve elements. The count is

(4 12) (4 8) (4 4) = \frac{12 !}{4 ! 4 ! 4 !} = \frac{12 !}{( 4 ! ) ^{3}}

, with no further division. Had the groups been regarded as indistinguishable one would divide by

3!

to get

\frac{12 !}{3 ! ( 4 ! ) ^{3}}

; because the problem labels them, the count without that factor is the one taken.

JEE Main · 2007 · 29 Apr 2007 · Offline · Single correct option