Question

The number of integral solutions for the equation $x+y+z+t=20,$ where $x,y,z,t$ are all $\ge -1,$ is

• A. ${}^{20}{C}_4$
• B. ${}^{23}{C}_3$
• C. ${}^{27}{C}_4$
• D. ${}^{27}{C}_3$

Answer 1

The correct option is D ${}^{27}{C}_3$

Given $x+y+z+t=20$
$x\ge -1,y\ge -1,z\ge -1$ and $t\ge -1$
Put $x+1=x_1,y+1=x_2,z+1=x_3,t+1=x_4$
$($where $x_1,x_2,x_3,x_4\ge 0)$
Hence given equation becomes $x_1+x_2+x_3+x_4=24$
$\therefore$ Number of non-negative integral solutions $={}^{\left(24+4-1\right)}{C}_{4-1}={}^{27}{C}_3$