Question

The number of integral solutions of the equation $x+y+z+t=20,$ such that $x\ge 0,y\ge 1,z\ge 2,t\ge 3,$ is

• A. $680$
• B. $720$
• C. $640$
• D. $560$

Answer 1

The correct option is A $680$

Let $X=x,Y=y-1,Z=z-2,T=t-3$ then $X+Y+Z+T=14$ where $X,Y,Z,T\ge 0$

$\therefore$Required number of solutions is number of non negative integral solutions for $X+Y+Z+T=14$ $={}^{14+4-1}{C}_{4-1}={}^{17}{C}_3=680$