Question

How many integral solutions are there for the equation ; $x+y+z+w=29$ when $x>1,y>2,z>3$ & $w\ge 0$ ?

Answer 1

$x+y+z+w=29$ ...(1)
$x>1,y>2,z>3,w\ge 0$
$\Rightarrow x-2\ge 0,y-3\ge 0,z-4\ge 0,w\ge 0$
let $x_1=x-2,x_2=y-3,x_3=z-4.$
$\Rightarrow x=x_1+2,y=x_2+3,z=x_3+4$
and then from (1).
$x_1+2+x_2+3+x_3+4+w=29$
$\Rightarrow x_1+x_2+x_3+w=20$
Hence total number of solution
${=}^{20+4-1}{C}_{4-1}{=}^{23}{C}_3=1771$