IZhO 2014 Problem 5

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 $\left(X_{1}, X_{2}, X_{3}, Y_{1}, Y_{2}, Y_{3}\right)$ of subsets of $U$, satisfying the following conditions

(i) $Y_{1} \subseteq X_{1} \subseteq U$ and $\left|X_{1}\right|=a$;

(ii) $Y_{2} \subseteq X_{2} \subseteq U \backslash Y_{1}$ and $\left|X_{2}\right|=b$;

(iii) $Y_{3} \subseteq X_{3} \subseteq U \backslash\left(Y_{1} \cup Y_{2}\right)$ and $\left|X_{3}\right|=c$.

Prove $f(\sigma(a), \sigma(b), \sigma(c))$ does not change, for permutations $\sigma$ of $a, b, c$.