Question 27
Let $U=\{1,2,3, \ldots, 2014\}$. For all $a, b, c \in \mathbb{N}$ let $f(a, b, c)$ be the number of ordered sextuplets $(X_{1}, X_{2}, X_{3}, Y_{1}, Y_{2}, Y_{3})$ of subsets of $U$, satisfying the following conditions
(i) $Y_{1} \subseteq X_{1} \subseteq U$ and $|X_{1}|=a$
(ii) $Y_{2} \subseteq X_{2} \subseteq U \setminus Y_{1}$ and $|X_{2}|=b$;
(iii) $Y_{3} \subseteq X_{3} \subseteq U \setminus (Y_{1} \cup Y_{2})$ and $|X_{3}|=c$.
Prove $f(\sigma(a), \sigma(b), \sigma(c))$ does not change, for permutations $\sigma$ of $a, b, c$.
(i) $Y_{1} \subseteq X_{1} \subseteq U$ and $|X_{1}|=a$
(ii) $Y_{2} \subseteq X_{2} \subseteq U \setminus Y_{1}$ and $|X_{2}|=b$;
(iii) $Y_{3} \subseteq X_{3} \subseteq U \setminus (Y_{1} \cup Y_{2})$ and $|X_{3}|=c$.
Prove $f(\sigma(a), \sigma(b), \sigma(c))$ does not change, for permutations $\sigma$ of $a, b, c$.