Unnamed Problem -2 (Combinatorics)

Given a set of numbers $N = \{1,2,\ldots,n\}$ and numbers $a, b, c |\; a + b = c$ and $a \leq b \leq c \leq n$

Let $A = \{a_1, a_2,.\ldots,a_{n \choose a}\;|\; a_1 \neq a_2 \neq\ldots\neq a_{n \choose a}\;;\;|a_x| = a\}$ ; $|A|$ = $n \choose a$

Similarly,

$B = \{b_1, b_2,.\ldots,b_{n \choose b}\;|\; b_1 \neq b_2 \neq\ldots\neq b_{n \choose b}\;;\;|b_x| = b\}$ ; $|B|$ = $n \choose b$ and

$C = \{c_1, c_2,.\ldots,c_{n \choose c}\;|\; c_1 \neq c_2 \neq\ldots\neq c_{n \choose c}\;;\;|c_x| = c\}$ ; $|C|$ = $n \choose c$

Now, consider a subset of C, $C^1 = \{c^x | c^x = a^x \cup b^x;\;c^x \in C, a^x \in A; b^x \in B\}$

The question is to find the minimal cardinality of the subset of A, such that $C^1 = C$

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Published by prasannar29