Question

Find the number of non-negative integral solutions of the equation $x+y+z=10.$

• A. ${}^{10}{C}_2$
• B. ${}^{12}{C}_2$
• C. ${}^{11}{C}_2$
• D. ${}^9{C}_2$

Answer 1

The correct option is B ${}^{12}{C}_2$

$\cdot$ The number of positive integral solutions of the equation $x_1+x_2+x_3+....+x_r=n$ is ${}^{n-1}{C}_{r-1}.$ $\cdot$ The number of non-negative integral solutions of the equation $x_1+x_2+x_3+....+x_r=n$ is ${}^{n+r-1}{C}_{r-1}.$

Required number of non-negative integral solution is ${=}^{10+3-1}{C}_{3-1}{=}^{12}{C}_2$