MathDB

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.

Problem Statement