PAMO 2008 Problem 3

Let $a,b,c$ be three positive integers such that $a<b<c$ . Consider the the sets $A,B,C$ and $X$ , defined as follows: $A=\{ 1,2,\ldots ,a \}$ , $B=\{a+1,a+2,\ldots,b\}$ , $C=\{b+1,b+2,\ldots ,c\}$ and $X=A\cup B\cup C$ . Determine, in terms of $a,b$ and $c$ , the number of ways of placing the elements of $X$ in three boxes such that there are $x,y$ and $z$ elements in the first, second and third box respectively, knowing that: i) $x\le y\le z$ ; ii) elements of $B$ cannot be put in the first box; iii) elements of $C$ cannot be put in the third box.